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Math 1351 011 September 7 2007 1 Announcements 0 Please write your Section number on your homework 750 Tuesday 1230 pm 751 Thursday 1230 pm 752 Tuesday 500 pm 753 Thursday 500 pm 0 Homework 3 will be posted today due next Friday 0 One volunteer note taker needed 0 Exam 1 on Friday 921 two weeks TTU Department of Mathematics amp Statistics Math 1351 011 September 7 2007 2 Graphing f 1 o If a b is a point on the graph of f then ba is a point on the graph of f l 0 You can graph f 1 by re ecting the graph of f about the line 16 y 0 Can think of it as graphing f ipping the paper then rotating 90O Clockwise TTU Department of Mathematics amp Statistics Math 1351 011 September 7 2007 3 Example of proving inverse trig identities Example For 1 g x g 1 ShOW that cossin1 x V1 962 Solution Let oz sin 10E Where g 04 g Solve With reference triangle TTU Department of Mathematics amp Statistics Math 1351 011 September 7 2007 4 Chapter 2 Limit of a function Consider a free falling body with no air resistance Falls approximately 8t 16152 feet in t seconds Previously we looked at average velocity over an interval of time Now we want to know instantaneous velocity at t 2 seconds We can express this as a limit We ll compute the average velocity over a smaller and smaller time interval near t 2 seconds Let 27 be the average velocity over the interval 19 g t g 2 distance traveled elapsed time 82 s19 2 19 1622 16192 01 624 fts TTU Department of Mathematics amp Statistics Math 1351 011 September 7 2007 5 We can make similar computations With smaller intervals from above or below t 2 Int 192 1992 19992 220001 22001 2291 Len 01 001 0001 00001 0001 001 v 624 6384 6398 640016 64016 6416 Average velocity seems to be approaching 64 as we make the time intervals smaller So we expect that the instantaneous velocity When t 2 Will be 64 TTU Department of Mathematics amp Statistics Math 1351 011 September 7 2007 6 Average velocity of the falling body over time interval 2 g t g 2 h is 82 h 82 162 m2 1622 2 h 2 h In this example the average velocity has a limiting value of 64 as the length h of the time interval tends to zero 2 2 hm w 64 h gtO h Informal limit de nition The notation lim f L means that the function values f can be made arbitrarily close to a unique number L by choosing x suf ciently close to c but not equal to 0 Notation Also written as fr gt L as x gt c TTU Department of Mathematics amp Statistics Math 1351 011 September 7 2007 7 Example Evaluate by table 2xx l 1 x 2 Llim ac gt0 E2 x 05 01 001 0001 0 0001 0005 fx 03431 02633 02513 02501 undef 02499 02494 Table pattern suggests that the limit L is 025 Be careful With calculators or computers and very small numbers Can introduce errors TTU Department of Mathematics amp Statistics Math 1351 011 September 7 2007 8 OneSided Limits Righthand limit lim L ac gtc if we can make f arbitrarily close to L by choosing x suf ciently close to c on a small interval 01 immediately to the right of c Lefthand limit lim L gtC if we can make f arbitrarily close to L by choosing x suf ciently close to c on a small interval a 0 immediately to the left of c TTU Department of Mathematics amp Statistics Math 1351 011 September 7 2007 9 Theorem onesided limit theorem The two sided limit limxmc f exists if and only if the tWO one sided limits exist and are equal Furthermore if lim f1cL lim x ac gtc ac gtc then lim L gtC TTU Department of Mathematics amp Statistics Math 1351 011 September 7 2007 10 Limits do not always exist If the limit of the function f fails to exist f is said to diverge asx gtc o The function may grow arbitrarily large or small as x gt c 1 Eg limxno 36 2 A function f that increases or decreases Without bound as x approaches 0 is said to tend to in nity inf as x gt 0 lim f inf if f increases Without bound gtC lim f inf if f decreases Without bound 1 gtC o The function may oscillate as x gt c 1 Eg limxgt0 sin 5 divergence by oscillation TTU Department of Mathematics amp Statistics COPYRIGHT NOTICE Adrian Banner The Calculus Lifesaver is published by Princeton University Press and copyrighted 2007 by Princeton University Press All rights reserved No part of this book may be reproduced in any form by any electronic or mechanical means including photocopying recording or information storage and retrieval without permission in writing from the publisher except for reading and browsing via the World Wide Web Users are not permitted to mount this le on any network servers Follow links for Class Use and other Permissions For more information send email to permissionspupressprincetonedu C H A l 39I E R l Functions Grophs and Lines Trying to do calculus without using functions would be one of the most point less things you could do If calculus had an ingredients list functions would be rst on it and by some margin too So the rst two chapters of this book are designed to jog your memory about the main features of functions This chapter contains a review of the following topics functions their domain codomain and range and the vertical line test inverse functions and the horizontal line test composition of functions odd and even functions graphs of linear functions and polynomials in general as well as a brief survey of graphs of rational functions exponentials and logarithms and 0 how to deal with absolute values Trigonometric functions or trig functions for short are dealt with in the next chapter So let s kick off with a review of what a function actually is Functions A function is a rule for transforming an object into another object The object you start with is called the input and comes from some set called the domain What you get back is called the output it comes from some set called the codomain Here are some examples of functions 0 Suppose you write 952 You have just de ned a function f which transforms any number into its square Since you didn t say what the domain or codomain are it s assumed that they are both R the set of all real numbers So you can square any real number and get a real number back For example f transforms 2 into 4 it transforms 712 into 14 and it transforms 1 into 1 This last one isn t much of a change at all but that s no problem the transformed object doesn t have to be different from the original one When you write 4 what you really mean 2 0 Functions Graphs and Lines is that f transforms 2 into 4 By the way f is the transformation rule while is the result of applying the transformation rule to the variable 95 So it s technically not correct to say fx is a function it should be f is a function77 Now let 995 x2 with domain consisting only of numbers greater than or equal to 0 Such numbers are called nonnegative This seems like the same function as f but it s not the domains are different For example f7l2 14 but 9712 isn t de ned The function 9 just chokes on anything not in the domain refusing even to touch it Since 9 and f have the same rule but the domain of g is smaller than the domain of f we say that g is formed by restricting the domain of 0 Still letting 952 what do you make of fhorse Obviously this is unde ned since you can t square a horse On the other hand let s set Mac number of legs 95 has where the domain of h is the set of all animals So hhorse 4 while hant 6 and hsalmon 0 The codomain could be the set of all nonnegative integers since animals don t have negative or fractional numbers of legs By the way what is M2 This isn t de ned of course since 2 isn t in the domain How many legs does a 2 have after all The question doesn t really make sense You might also think that hchair 4 since most chairs have four legs but that doesn t work either since a chair isn t an animal and so chair is not in the domain of ii That is hchair is unde ned 0 Suppose you have a dog called Junkster Unfortunately poor Junkster has indigestion He eats something then chews on it for a while and tries to digest it fails and hurls Junkster has transformed the food into i i something else altogether We could let color of barf when Junkster eats x where the domain of j is the set of foods that Junkster will eat The codomain is the set of all colors For this to work we have to be con dent that whenever Junkster eats a taco his barf is always the same color say red If it s sometimes red and sometimes green that s no good a function must assign a unique output for each valid input Now we have to look at the concept of the range of a function The range is the set of all outputs that could possibly occur You can think of the function working on transforming everything in the domain one object at a time the collection of transformed objects is the range You might get duplicates but that s OK So why isn t the range the same thing as the codomain Well the range is actually a subset of the codomain The codomain is a set of possible outputs while the range is the set of actual outputs Here are the ranges of the functions we looked at above NH Section 111 Interval notation o 3 o If x2 with domain R and codomain R the range is the set of nonnegative numbers After all when you square a number the result cannot be negative How do you know the range is all the nonnegative numbers Well if you square every number you de nitely cover all nonnegative numbers For example you get 2 by squaring 2 or o If 995 952 where the domain of g is only the nonnegative numbers but the codomain is still all of R the range will again be the set of nonnegative numbers When you square every nonnegative number you still cover all the nonnegative numbers If Mac is the number of legs the animal as has then the range is all the possible numbers of legs that any animal can have I can think of animals that have 0 2 4 6 and 8 legs as well as some creepycrawlies with more legs If you include individual animals which have lost one or more legs you can also include 1 3 5 and 7 in the mix as well as other possibilities In any case the range of this function isn t so clearcut you probably have to be a biologist to know the real answer Finally if is the color of Junkster s barf when he eats x then the range consists of all possible barfcolors I dread to think what these are but probably bright blue isn t among them Wewol notation In the rest of this book our functions will always have codomain R and the domain will always be as much of R as possible unless stated otherwise So we ll often be dealing with subsets of the real line especially connected intervals such as x 2 S x lt 5 It s a bit of a pain to write out the full set notation like this but it sure beats having to say all the numbers between 2 and 5 including 2 but not 577 We can do even better using interval notation We ll write 1 b to mean the set of all numbers between a and b including a and b themselves So 11 means the set of all x such that a S x S b For example 2 5 is the set of all real numbers between 2 and 5 including 2 and 5 It s not just the set consisting of 2 3 4 and 5 don t forget that there are loads of fractions and irrational numbers between 2 and 5 such as 52 W and 7L An interval such as 1 b is called closed If you don t want the endpoints change the square brackets to parentheses In particular a b is the set of all numbers between a and b not including a or I So if x is in the interval a b we know that a lt x lt b The set 2 5 includes all real numbers between 2 and 5 but not 2 or 5 An interval of the form a b is called open You can mix and match 1 b consists of all numbers between a and b including a but not I And a b includes b but not 1 These intervals are closed at one end and open at the other Sometimes such intervals are called halfopen An example is the set x 2 S x lt 5 from above which can also be written as 2 5 There s also the useful notation a 0 for all the numbers greater than a not including a 1 0 is the same thing but with 1 included There are three other possibilities which involve 700 all in all the situation looks like this A 0 Functions Graphs and Lines 11 Ialtzltb 4 o Z gt 11 LnaSsz lt o o gt a 1 ab Ialtz b 4 o z gt ab Ia zltb lt o o gt a 1 1100 IIIgt1 lt g gt 100 112a lt gt 7001 III ltb lt o gt b 7oob 11 Sb 4 Z gt 0000 R 4 H 2 Finding the domain Sometimes the de nition of a function will include the domain This was the case for example with our function g from Section 11 above Most of the time however the domain is not provided The basic convention is that the domain consists of as much of the set of real numbers as possible For example if Mac the domain can t be all of R since you can t take the square root of a negative number The domain must be 0 0 which is just the set of all numbers greater than or equal to 0 OK so square roots of negative numbers are bad What else can cause a screwup Here s a list of the three most common possibilities 1 The denominator of a fraction can t be zero 2 You can t take the square root or fourth root sixth root and so on of a negative number 3 You can t take the logarithm of a negative number or of 0 Remember logs If not see Chapter 9 You might recall that tan90 is also a problem but this is really a special case of the rst item above You see sin90 1 tanmoo cos90 6 so the reason tan90 is unde ned is really that a hidden denominator is zero Here s another example if we try to de ne 7 log10x 826 7 295 7 x 7 2x 19 then what is the domain of f Well for to make sense here s what needs to happen x 0 We need to take the square root of 26 7 295 so this quantity had better be nonnegative That is 26 7 2x 2 0 This can be rewritten as x S 13 iii 0 Section 113 Finding the range using the graph 0 5 0 We also need to take the logarithm of x 8 so this quantity needs to be positive Notice the difference between logs and square roots you can take the square root of 0 but you can t take the log of 0 Anyway we need as 8 gt 0 so as gt 78 So far we know that 78 lt x S 13 so the domain is at most 78 13 o The denominator can t be 0 this means that 9572 y 0 and 9519 y 0 In other words as y 2 and x y 719 This last one isn t a problem since we already know that x lies in 78 13 so x can t possibly be 719 We do have to exclude 2 though So we have found that the domain is the set 78 13 except for the number 2 This set could be written as 78132i Here the backslash means not including77 Finding ine ignge using ine gigpn Let s de ne a new function F by specifying that its domain is 72 1 and that Fx x2 on this domain Remember the codomain of any function we look at will always be the set of all real numbers ls F the same function as f where x2 for all real numbers as The answer is no since the two functions have different domains even though they have the same rule As in the case of the function g from Section 11 above the function F is formed by restricting the domain of Now what is the range of F Well what happens if you square every number between 72 and 1 inclusive You should be able to work this out directly but this is a good opportunity to see how to use a graph to nd the range of a function The idea is to sketch the graph of the function then imagine two rows of lights shining from the far left and far right of the graph horizontally toward the yaxis The curve will cast two shadows one on the left side and one on the right side of the yaxis The range is the union of both shadows that is if any point on the yaxis lies in either the lefthand or the righthand shadow it is in the range of the function Let s see how this works with our function F gt 4 E lt 3 gt O lt quotCS 6 lt m 3 0 Functions Graphs and Lines HA The lefthand shadow covers all the points on the yaxis between 0 and 4 inclusive which is 04 on the other hand the righthand shadow covers the points between 0 and 1 inclusive which is 0 1 The righthand shadow doesn t contribute anything extra the total coverage is still 04 This is the range of F The vertical Hoe test In the last section we used the graph of a function to nd its range The graph of a function is very important it really shows you what the function looks like77 We ll be looking at techniques for sketching graphs in Chapter 12 but for now I d like to remind you about the vertical line test You can draw any gure you like on a coordinate plane but the result may not be the graph of a function So what s special about the graph of a function What is the graph of a function f anyway Well it s the collection of all points with coordinates x where x is in the domain of Here s another way of looking at this start with some number x If x is in the domain you plot the point x which of course is at a height of units above the point x on the xaxis If 95 isn t in the domain you don t plot anything Now repeat for every real number x to build up the graph Here s the key idea you can t have two points with the same xcoordinate In other words no two points on the graph can lie on the same vertical line Otherwise how would you know which of the two or more heights above the point x on the xaxis corresponds to the value of So this leads us to the vertical line test if you have some graph and you want to know whether it s the graph of a function see whether any vertical line intersects the graph more than once If so it s not the graph of a function but if no vertical line intersects the graph more than once you are indeed dealing with the graph of a function For example the circle of radius 3 units centered at the origin has a graph like this Such a commonplace object should be a function right No check the vertical lines that are shown in the diagram Sure to the left of 73 or to the right of 3 there s no problemithe vertical lines don t even hit the graph which is ne Even at 73 or 3 the vertical lines only intersect the curve in one point each which is also ne The problem is when x is in the interval 73 3 For Section 12 Inverse Functions 0 7 any of these values of x the vertical line through as 0 intersects the circle twice which screws up the circle s potential functionstatus You just don t know whether is the top point or the bottom point The best way to salvage the situation is to chop the circle in half hori zontally and choose only the top or the bottom half The equation for the whole circle is x2 y2 9 whereas the equation for the top semicircle is y V9 7 952 The bottom semicircle has equation y ixQ 7 952 These last two are functions both with domain 73 3 If you felt like chopping in a different way you wouldn t actually have to take semicirclesiyou could chop and change between the upper and lower semicircles as long as you don t vi olate the vertical line test For example here s the graph of a function which also has domain 73 3 The vertical line test checks out so this is indeed the graph of a function loverse Functions Let s say you have a function You present it with an input as provided that x is in the domain of f you get back an output which we call Now we try to do things all backward and ask this question if you pick a number 3 what input can you give to f in order to get back y as your output Here s how to state the problem in mathspeak given a number y what x in the domain of f satis es y The rst thing to notice is that y has to be in the range of Otherwise by de nition there are no values of x such that 3 There would be nothing in the domain that f would transform into 3 since the range is all the possible outputs On the other hand if y is in the range there might be many values that work For example if x2 with domain R and we ask what value of x transforms into 64 there are obviously two values of x 8 and 78 On the other hand if 995 x3 and we ask the same question there s only one value of x which is 4 The same would be true for any number we give to g to transform because any number has only one real cube root So here s the situation we re given a function f and we pick y in the range of Ideally there will be exactly one value of x which satis es 3 If this is true for every value of y in the range then we can de ne a new 8 0 Functions Graphs and Lines l2 4 function which reverses the transformation Starting with the output y the new function nds the one and only input as which leads to the output The new function is called the inverse function off and is written as fill Here s a summary of the situation in mathematical language H Start with a function f such that for any y in the range of f there is exactly one number x such that y That is different inputs give 71 different outputs Now we will de ne the inverse function f m i The domain of f 1 is the same as the range of 9quot The range of f 1 is the same as the domain of The value of f 1y is the number x such that y So if fxy then f 1y The transformation f 1 acts like an undo button for f if you start with x and transform it into y using the function f then you can undo the effect of the transformation by using the inverse function f 1 on y to get as back This raises some questions how do you see if there s only one value of x that satis es the equation y If so how do you nd the inverse and what does its graph look like If not how do you salvage the situation We ll answer these questions in the next three sections th The horizoofol Hoe fesf For the rst questionihow to see that there s only one value of x that works for any 3 in the rangeiperhaps the best way is to look at the graph of your function We want to pick y in the range of f and hopefully only have one value of x such that y What this means is that the horizontal line through the point 031 should intersect the graph exactly once at some point x That as is the one we want the horizontal line intersects the curve more than once there would be multiple potential inverses x which is bad In that case the only way to get an inverse function is to restrict the domain we ll come back to this very shortly What if the horizontal line doesn t intersect the curve at all Then y isn t in the range after all which is OK So we have just described the horizontal line test if every horizontal line intersects the graph of a function at most once the function has an inverse If even one horizontal line intersects the graph more than once there isn t an inverse function For example look at the graphs of x3 and 995 952 91 x i2 i2 M 0 Section 122 Finding the inverse 0 9 No horizontal line hits y more than once so f has an inverse On the other hand some of the horizontal lines hit the curve y 995 twice so 9 has no inverse Here s the problem if you want to solve y x2 for x where y is positive then there are two solutions as and x i You don t know which one to take Finding ine inverse Now let s move on to the second of our questions how do you nd the inverse of a function f Well you write down y and try to solve for x In our example of 953 we have y 953 so as This means that f 1y If the variable y here offends you by all means switch it to x you can write f 1x 3 if you prefer Of course solving for x is not always easy and in fact is often impossible On the other hand if you know what the graph of your function looks like the graph of the inverse function is easy to nd The idea is to draw the line y x on the graph then pretend that this line is a twosided mirror The inverse function is the re ection of the original function in this mirror When x3 here s what f 1 looks like The original function f is re ected in the mirror y x to get the inverse function Note that the domain and range of both f and f 1 are the whole real line Restricting ine domain Finally we ll address our third question if the horizontal line test fails and there s no inverse what can be done Our problem is that there are multiple values of x that give the same y The only way to get around the problem is to throw away all but one of these values of 95 That is we have to decide which one of our values of x we want to keep and throw the rest away As we saw in Section 11 above this is called restricting the domain of our function Effectively we ghost out part of the curve so that what s left no longer fails the horizontal line test For example if 995 952 we can ghost out the left half of the graph like this l0 0 Functions Graphs and Lines The new unghosted curve has the reduced domain 0 gt0 and satis es the horizontal line test so there is an inverse function More precisely the function h which has domain 0 gt0 and is de ned by Mac x2 on this domain has an inverse Let s play the re ection game to see what it looks like To nd the equation of the inverse we have to solve for x in the equation y x2 Clearly the solution is x or x i but which one do we need We know that the range of the inverse function is the same as the domain of the original function which we have restricted to be 0 So we need a nonnegative number as our answer and that has to be ac y That is h 1y Of course we could have ghosted out the right half of the original graph to restrict the domain to 700 0 In that case we d get a function j which has domain 700 0 and again satis es 952 but only on this domain This function also has an inverse but the inverse is now the negative square root j 1y i By the way if you take the original function 9 given by 995 x2 with domain 700 0 which fails the horizontal line test and try to re ect it in the mirror y as you get the following picture i211 Section 124 lnverses of inverse functions 0 l l Notice that the graph fails the vertical line test so it s not the graph of a function This illustrates the connection between the vertical and horizontal line testsiwhen horizontal lines are re ected in the mirror y as they become vertical lines inverses of inverse funoiions One more thing about inverse functions if f has an inverse it s true that x for all x in the domain of f and also that y for all y in the range of Remember the range of f is the same as the domain of f l so you can indeed take f 1y for y in the range of f without causing any screwupsi For example if 953 then f has an inverse given by f 1x 35 and so W x for any 95 Remember the inverse function is like an undo button We use as as an input to f and then give the output to f l this undoes the transformation and gives us back as the original number Similarly y So 1quot1 is the inverse function of f and f is the inverse function of f li In other words the inverse of the inverse is the original function Now you have to be careful in the case where you restrict the domain Let 995 x2 we ve seen that you need to restrict the domain to get an inverse Let s say we restrict the domain to 0 gt0 and carelessly continue to refer to the function as 9 instead of h as in the previous section We would then say that g 1x If you calculate 99 1x you nd that this is y which equals 95 provided that x 2 0 Otherwise you can t take the square root in the rst place On the other hand if you work out g 1gx you get xxj which is not always the same thing as as For example i x 72 t en 952 4 an so V xZl 2 So it s not true in general that g 1gx x The problem is that 72 isn t in the restricteddomain version of g Technically you can t even compute 972 since 72 is no longer in the domain of 9 We really should be working with 1 not 9 so that we remember to be more careful Nevertheless in practice mathematicians will often restrict the domain with out changing lettersl So it will be useful to summarize the situation as follows If the domain of a function f can be restricted so that f has an inverse f l then 0 y for all y in the range of f but 0 may not equal as in fact as only when x is in the restricted domain We ll be revisiting these important points in the context of inverse trig func tions in Section 1026 of Chapter 10 Composiiion oi Funoiions Let s say we have a function 9 given by 995 952 You can replace x by anything you like as long as it makes sense For example you can write l2 0 Functions Graphs and Lines gy 32 or 995 5 x 5 This last example shows that you need to be very careful with parentheses It would be wrong to write gx5 x52 since this is just x 25 which is not the same thing as x 5 If in doubt use parentheses That is if you need to write out fsomething replace every instance of x by something making sure to include the parentheses Just about the only time you don t need to use parentheses is when the function is an exponential functionifor example if Mac 3 then you can just write M952 6 312 You don t need parentheses since you re already writing the x2 6 as a superscript Now consider the function f de ned by cosx2 lfl give you a number as how do you compute Well rst you square it then you take the cosine of the result Since we can decompose the action of into these two separate actions which are performed one after the other we might as well describe those actions as functions themselves So let 995 x2 and Mac cosx To simulate what f does when you use as as an input you could rst give as to g to square it and then instead of taking the result back you could ask 9 to give its result to 1 instead Then 1 spits out a number which is the nal answer The answer will of course be the cosine of what came out of g which was the square of the original as This behavior exactly mimics f so we can write Another way of expressing this is to write f h o 9 here the circle means composed with77 That is f is h composed with g or in other words f is the composition of h and 9 What s tricky is that you write 1 before 9 reading from left to right as usual but you apply 9 rst I agree that it s confusing but what can I sayiyou just have to deal with it It s useful to practice composing two or more functions together For example if 9x 2 Mac 5x4 and 2x 7 1 what is a formula for the function f g o h 0 j Well just replace one thing at a time starting with j then 1 then 9 So x 901006 901296 71 95096 14 2521 U4 You should also practice reversing the process For example suppose you start off with l 7 tan5 log2x How would you decompose f into simpler functions Zoom in to where you see the quantity as The rst thing you do is add 3 so let 995 x 3 Then you have to take the base 2 logarithm of the resulting quantity so set Mac log2x Next multiply by 5 so set 595 Then take the tangent so put Mac Finally take reciprocals so let 71195 195 With all these de nitions you should check that x mkjh9x Using the composition notation you can write fmokojohog Section 13 Composition of Functions 0 This isn t the only way to break down For example we could have combined 1 and j into another function n where 5log2 Then you should check that n j o h and f m o k 0 n o 9 Perhaps the original decomposition involvingj and h is better because it breaks down f into more elementary steps but the second one involving n isn t wrong After all 5log2x is still a pretty simple function of x Beware composition of functions isn t the same thing as multiplying them together For example if x2 sinx then f is not the composition of two functions To calculate f for any given as you actually have to nd both 952 and sinx it doesn t matter which one you nd rst unlike with composition and then multiply these two things together If 995 x2 and Mac sinx then we d write gxhx or f gh Compare this to the composition of the two functions j g o h which is given by NC 90166 9sinx 8111062 or simply sin2x The function j is a completely different function from the product 952 sinx It s also different from the function k h o g which is also a composition of g and h but in the other order Mac hgx M952 sinx2 This is yet another completely different function The moral of the story is that products and compositions are not the same thing and furthermore the order of the functions matters when you compose them but not when you multiply them together One simple but important example of composition of functions occurs when you compose some function f with 995 x 7 a where a is some constant number You end up with a new function 1 given by Mac fx7 1 A useful point to note is that the graph of y Mac is the same as the graph of y except that it s shifted over 1 units to the right If a is negative then the shift is to the left The way to think of this for example is that a shift of 73 units to the right is the same as a shift of 3 units to the left So how would you sketch the graph of y x 7 If This is the same as y 952 but with x replaced by x 7 1 So the graph of y 952 needs to be shifted to the right by 1 unit and looks like this yxilgt2 l4 0 Functions Graphs and Lines Similarly the graph of y x 22 is the graph of y x2 shifted to the left by 2 units since you can interpret as 2 as x 7 72 Odd and Even Functions Some functions have some symmetry properties that make them easier to deal with Consider the function f given by x2 Pick any positive number you like I ll choose 3 and hit it with f I get 9 Now take the negative of that number 73 in my case and hit that with f I get 9 again You should get the same answer both times as I did regardless of which number you chose You can express this phenomenon by writing for all 95 That is if you give as to f as an input you get back the same answer as if you used the input 795 instead Notice that 995 x4 and Mac 956 also have this propertyiin fact x where n is any even number 71 could in fact be negative has the same property Inspired by this we say that a function f is even if for all x in the domain of It s not good enough for this equation to be true for some values of x it has to be true for all x in the domain of Now let s say we play the same game with 953 Take your favorite positive number I ll stick with 3 and hit that with f I get 27 Now try again with the negative of your number 73 in my case I get 727 and you should also get the negative of what you got before You can express this mathematically as Once again the same property holds for jx as when n is any odd number and once again 71 could be negative So we say that a function f is add if for all x in the domain of In general a function might be odd it might be even or it might be neither odd nor even Don t forget this last point Most functions are neither odd nor even On the other hand there s only one function that s both odd and even which is the rather boring function given by 0 for all as we ll call this the zero function Why is this the only odd and even function Let s convince ourselves If the function f is even then for all 95 But if it s also odd then for all 95 Take the rst of these equations and subtract the second from it You should get 0 2fx which means that 0 This is true for all as so the function f must just be the zero function One other nice observation is that if a function f is odd and the number 0 is in its domain then 0 Why is it so Because is true for all x in the domain of f so let s try it for x 0 You get f70 7f0 But 70 is the same thing as 0 so we have f0 7f0i This simpli es to 2f0 0 or 0 as claimed Anyway starting with a function f how can you tell if it is odd even or neither And so what if it is odd or even anyway Let s look at this second question before coming back to the rst one One nice thing about knowing that a function is odd or even is that it s easier to graph the function In fact if you can graph the righthand half of the function the lefthand half is a piece of cake Let s say that f is an even function Then since f7x the graph of y is at the same height above the xcoordinates x and 7x This is true for all as so the situation looks something like this Section 14 Odd and Even Functions 0 Same height We can conclude that the graph of an even function has mirror sym metry about the yaxis So if you graph the right half of a function which you know is even you can get the left half by re ecting the right half about the yaxis Check the graph of y x2 to make sure that it has this mirror symmetry On the other hand let s say that f is an odd function Since we have ier 7fx the graph of y is at the same height above the xcoordinate x as it is below the xcoordinate 795 Of course if is negative then you have to switch the words above and below7 In any case the picture looks like this Same length opposite signs The symmetry is now a point symmetry about the origin That is the graph of an odd function has 180 point symmetry about the origin This means that if you only have the right half of a function which you know is odd you can get the left half as follows Pretend that the curve is sitting on top of the paper so you can pick it up if you like but you can t change its shape Instead of picking it up put a pin through the curve at the origin remember odd functions must pass through the origin if they are de ned at 0 and then spin the whole curve around half a revolution This is what the lefthand half of the graph looks like This doesn t work so well if the curve isn t continuous that is if the curve isn t all in one piece Check to see that the above graph and also the graph of y 953 have this symmetry Now suppose f is de ned by the equation log52x676x23i How do you tell if f is odd even or neither The technique is to calculate by replacing every instance of x with 7x making sure not to forget the parentheses around 795 and then simplifying the result If you end up with the original expression then f is even if you end up with the negative of the original expression f7x then f is odd if you end up with a mess that isn t either or 7fx then f is neither or you didn t simplify enoughl l6 0 Functions Graphs and Lines In the example above you d write fx10g527x6 7 6H 3 Iog5lt2z6 7 6x2 3 which is actually equal to the original f So the function f is even How about 2x3 95 2953 95 71 7 995 3962 5 and Mac 3962 5 Well for 9 we have 7 27x3 7x 7 72x3 7 x 9 7x 7 Mix 5 7 3952 5 Now you have to observe that you can take the minus sign out front and write 2x3x 3x25 9796 which you notice is equal to 79x That is apart from the minus sign we get the original function back So 9 is an odd function How about 1 We have 27x3 7x71 72x3 7 x 71 h 7 96 Mix 5 3x2 5 Once again we take out the minus sign to get 72953 x 1 M x 7 3x2 5 Hmm this doesn t appear to be the negative of the original function because of the 1 term in the numerator It s not the original function either so the function h is neither odd nor even Let s look at one more example Suppose you want to prove that the product of two odd functions is always an even function How would you go about doing this Well it helps to have names for things so let s say we have two odd functions f and 9 We need to look at the product of these functions so let s call the product 1 That is we de ne Mac So our task is to show that h is even We ll do this by showing that Mix Mac as usual It will be helpful to note that and 9795 7995 since f and g are odd Let s start with Mix Since 1 is the product of f and 9 we have Mix f7xg7xi Now we use the oddness of f and g to express this last term as The minus signs now come out front and cancel out so this is the same thing as which of course equals We could and should express all this text mathematically like this Mix f96996 f 990 fx996gt M96 Anyway since Mix Mac the function h is even Now you should try to prove that the product of two even functions is always even and also that the product of an odd and an even function must be odd Go on do it now Section 15 Graphs of Linear Functions 0 l 7 Graphs of Linear Functions Functions of the form mac 1 are called linear There s a good reason for this the graphs of these functions are lines As far as we re concerned the word line always means straight line 77 The slope of the line is given by m Imagine for a moment that you are in the page climbing the line as if it were a mountain You start at the left side of the page and head to the right like this If the slope m is positive as it is in the above picture then you are heading uphill The bigger m is the steeper the climb On the other hand if the slope is negative then you are heading downhill The more negative the slope the steeper the downhill grade If the slope is zero then the line is at or horizontaliyoulre going neither uphill nor downhill just trudging along a at line To sketch the graph of a linear function you only need to identify two points on the graph This is because there s only one line that goes through two different points You just put your ruler on the points and draw the line One point is easy to nd namely the yintercept Set as 0 in the equation 3 mac b and you see that y m X 0 b b That is the yintercept is equal to b so the line goes through 0 b To nd another point you could nd the xintercept by setting y 0 and nding what 95 is This works pretty well except in two cases The rst case is when I 0 in which case we are just dealing with y mm This goes through the origin so the xintercept and the yintercept are both zero To get another point you ll just have to substitute in x 1 and see that y m So the line 3 mac goes through the origin and 1 For example the line 3 7295 goes through the origin and also through 1 72 so it looks like this l8 0 Functions Graphs and Lines The other bad case is when m 0 But then we just have y b which is a horizontal line through 0 b or a more interesting example consider y x 7 1 The yintercept is 71 and the slope is To sketch the line nd the xintercept by setting y 0 We get 0 x 7 1 which simpli es to x 2 So the line looks like this Now let s suppose you know that you have a line in the plane but you don t know its equation If you know it goes through a certain point and you know what its slope is then you can nd the equation of the line You really really really need to know how to do this since it comes up a lot This formula called the pointslope form of a linear function is what you need to know If a line goes through 950 yo and has slope m then its equation is y 7 yo m x 7 x0 For example what is the equation of the line through 72 5 which has slope 73 It is y 7 5 7395 7 72 which you can expand and simplify down to y 73x 7 1 Sometimes you don t know the slope of the line but you do know two points that it goes through How do you nd the equation The technique is to nd the slope then use the preVious idea with one of the points your choice to nd the equation First you need to know this y27y1 9627961 If a line goes through x1 y1 and x2 yg its slope is equal to So what is the equation of the line through 73 4 and 2 76 Let s nd the slope rst 76 7 4 710 slope7 27790 7 5 7 2 We now know that the line goes through 734 and has slope 72 so its equation is y 7 4 7295 7 73 or after simplifying y 72x 7 2 Alterna tively we could have used the other point 2 76 with slope 72 to see that the equation of the line is y 7 76 7295 7 2 which simpli es to y 72x 7 2 Thankfully this is the same equation as before7it doesn t matter which point you pick as long as you have used both points to nd the slope Section 16 Common Functions and Graphs o l 9 Common Functions and Graphs Here are the most important functions you should know about 1 Polynomials these are functions built out of nonnegative integer powers of x You start with the building blocks 1 x x2 x3 and so on and you are allowed to multiply these basic functions by numbers and add a nite number of them together For example the polynomial 5x474x310 is formed by taking 5 times the building block 954 and 74 times the building block 953 and 10 times the building block 1 and adding them together You might also want to include the intermediate building blocks 952 and x but since they don t appear you need to take 0 times of each The amount that you multiply the building block as by is called the coe icient of as For example in the polynomial f above the coefficient of x4 is 5 the coefficient of x3 is 74 the coefficients of x2 and x are both 0 and the coefficient of 1 is 10 Why allow 95 and 1 by the way They seem different from the other blocks but they re not really as x1 and 1 950 The highest number n such that x has a nonzero coefficient is called the degree of the polynomial For example the degree of the above polynomial f is 4 since no power of as greater than 4 is present The mathematical way to write a general polynomial of degree n is P06 11796quot an71x 1 12952 1195 10 where 17 is the coefficient of x a 1 is the coefficient of xn l and so on down to 10 which is the coefficient 0 1 Since the functions as are the building blocks of all polynomials you should know what their graphs look like The even powers mostly look similar to each other and the same can be said for the odd powers Here s what the graphs look like from 950 up to 957 20 0 Functions Graphs and Lines Sketching the graphs of more general polynomials is more difficult Even nd ing the xintercepts is often impossible unless the polynomial is very simple There is one aspect of the graph that is fairly straightforward which is what happens at the far left and right sides of the graph This is determined by the socalled leading coe icient which is the coef cient of the highestdegree term This is basically the number 17 de ned above For example in our polynomial 5x4 7 4x3 10 from above the leading coef cient is 5 In fact it only matters whether the leading coef cient is positive or negative It also matters whether the degree of the polynomial is odd or even so there are four possibilities for what the edges of the graph can look like n even an gt 0 n odd an gt 0 n even an lt 0 n odd an lt 0 The wiggles in the center of these diagrams aren t relevant7they depend on the other terms of the polynomial The diagram is just supposed to show what the graphs look like near the left and right edges In this sense the graph of our polynomial 5x4 7 4x3 10 looks like the leftmost picture above since n 4 is even and a7 5 is positive Let s spend a little time on degree 2 polynomials which are called quadrat ics Instead of writingpx a2x2a1xa0 it s easier to write the coef cients as a b and c so we have px 1952 bx c Quadratics have two one or zero real roots depending on the sign of the discriminant The discrimi nant which is often written as A is given by A b2 7 4ac There are three possibilities If A gt 0 then there are two roots if A 0 there is one root which is called a double mat and if A lt 0 then there are no roots In the rst two cases the roots are given by 7b i Vb 7 4ac 2a Notice that the expression in the square root is just the discriminant An im portant technique for dealing with quadratics is completing the square Here s how it works We ll use the example of the quadratic 2x2 7 3x 10 The rst step is to take out the leading coef cient as a factor So our quadratic becomes 2952 7 95 5 This reduces the situation to dealing with a manic quadratic which is a quadratic with leading coef cient equal to 1 So let s worry about 952 7 3x 5 The main technique now is to take the coef cient of ac which in our example is 7 divide it by 2 to get 7 and square it We get We wish that the constant term were instead of 5 so let s do some Section 16 Common Functions and Graphs o mental gymnastics I27 I5ix27 3573 2 7 2 16 16 Why on earth would we want to add and subtract 1796 Because the rst three terms combine to form x 7 3 So we have 27 57 27 3 5737 7 2573 9 2x 7x 2x 16 16 x4 16 Now we just have to work out the last little bit which is just arithmetic 5 7 E Putting it all together and restoring the factor of 2 we have 3 3 2 71 2 7 2 7 7 7 7 7 7 2x 3x10 2ltx 2x5gt 296 4 16 3 2 71 2 7 7 7 lt9 4 8 It turns out that this is a much nicer form to deal with in a number of situa tions Make sure you know how to complete the square since we ll be using this technique a lot in Chapters 18 and 19 2 Rational functions these are functions of the form WU 1196 where p and q are polynomials Rational functions will pop up in many different contexts and the graphs can look very different depending on the polynomials p and q The simplest examples of rational functions are poly nomials themselves which arise when 1196 is the constant polynomial l The next simplest examples are the functions 196 where n is a positive integer Let s look at some of the graphs of these functions The odd powers look similar to each other and the even powers look similar to each other It s worth knowing what these graphs look like 22 0 Functions Graphs and Lines 3 Exponentials and logarithms you need to know What graphs of expo nentials look like For example here is y 2 The graph of y bx for any other base I gt 1 looks similar to this Things to notice are that the domain is the Whole real line the yintercept is l the range is 0 gt0 and there is a horizontal asymptote on the left at y 0 In particular the curve y bx does not I repeat not touch the xaxis no matter What it looks like on your graphing calculator We ll be looking at asymptotes again in Chapter 3 The graph of y 2 is just the re ection of y 2 in the yaxis How about When the base is less than 1 For example consider the graph of y Notice that 12 2 so the above graph of y 2 is also the graph of y ax since 2 and are equal for any x The same sort of thing happens for y b for any 0 lt b lt 1 not just I Now notice that the graph of y 2 satis es the horizontal line test so there is an inverse function This is in fact the base 2 logarithm Which is Written y log2xi Using the line y x as a mirror the graph ofy log2x looks like this Section 16 Common Functions and Graphs o z 1 mirror y I y 10g21 The domain is 0 0 note that this backs up what I said earlier about not being able to take logarithms of a negative number or of 0 The range is all of 700 0 and there s a vertical asymptote at x 0 The graphs of log10x and indeed logbx for any I gt 1 are very similar to this one The log func tion is very important in calculus so you should really know how to draw the above graphi We ll look at other properties of logarithms in Chapter 9 4 Trig functions these are so important that the entire next chapter is devoted to themi 5 Functions involving absolute values let s take a close look at the absolute value function f given by Here s the de nition of Another way of looking at is that it is the distance between as and 0 on the number line More generally you should learn this nice fact lac 7 yl is the distance between as and y on the number linei For example suppose that you need to identify the region lac 7 ll 3 3 on the number line You can interpret the inequality as the distance between as and l is less than or equal to 3 77 That is we are looking for all the points that are no more than 3 units away from the number 1 So let s take a number line and mark in the number 1 as follows ltgt l The points which are no more than 3 units away extend to 72 on the left and 4 on the right so the region we want looks like this 3 units 3 units W 72 l 4 So the region lac 7 ll 3 3 can also be described as 72 4 24 0 Functions Graphs and Lines It s also true that To check this suppose that x 2 0 then xxi x no problem If instead x lt 0 then it can t be true that xQT 95 since the lefthand side is positive but the righthand side is negative The correct equation is x 795 now the righthand side is positive since it s minus a negative number If you look back at the de nition of M you ll see that we have just proved that Even so to deal with M it s much better to use the piecewise de nition than to write it as Finally let s take a look at some graphs If you know what the graph of a function looks like you can get the graph of the absolute value of that function by re ecting everything below the xaxis up to above the xaxis using the xaxis as your mirror For example here s the graph of y M which comes from re ecting the bottom portion of y x in the xaxis yl1l mirror graxis How about the graph of y llog2 Using the re ection of the graph of y log2x above this is what the absolute value version looks like 2 llogzml mirror graxis Anyway that s alll have to say about functions apart from trig functions which are the subject of the next chapter Hopefully you ve seen a lot of the stuff in this chapter before Most of the material in this chapter is used over and over again in calculus so make sure you really get on top of it all as soon as you can