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Date Created: 09/19/15
Math 1330 ONLINE Week 1 Session Notes Instructor Marjorie Marks Email mmarkscmathuhedu Course Website onlinemathuheducourses Click on the link to Math 1314 CASA Website casauhedu My personal website mathuhedummarksc Open Popper popl at casauhedu An Introduction to Functions Most of this course will deal with functions Suppose we start with two sets A and B A function is a rule which assigns one and only one element of set B to each element in set A SetA is called the domain ofthe function and set B is called the range We ll start by looking at mappings A mapping relates each element in the oval on the left with an element in the oval on the right You need to be able to state whether or not the mapping de nes a function and if it de nes a function you should be able to state the domain and range of the function Example 1 State whether or not the mapping represents a function If it does identify its domain and range b 5 c 7 6 A B Functions are usually written using function notation If an equation is solved for y such as y mx b we would write this using function notation as f x mx b read fof x denoting the value of the function at x We can also use other lower case letters to denote a function such as g h39 k etc Often you will be asked to state the domain of a stated function Domain is a subset of the set of real numbers First we ll review interval notation Reminder Interval Notation 3 5 all x such that 3 lt x lt 5 3 5 all x such that 3 S x S 5 3 5 all x such that 3 S x lt 5 3 00 all x such that x 2 3 00 5 all x such that x lt 5 oo 00 all real numbers Example 2 Express using interval notation 3 lt x S 5 Example 3 Express using interval notation x lt 3 Example 4 Express using interval notation x 72 7 Example 5 Is x2 y2 25 a function Why or why not If it is a function state the domain and range using interval notation Example 6 Is x2 y 1 a function Why or why not If it is a function state the domain and range using interval notation Next we ll state the domain of several types of functions The domain of any polynomial function is 0000 or all real numbers The domain of any rational function where both the numerator and the denominator are polynomials is all real numbers except the values of x for which the denominator equals The domain of any radical function with even index is the set of real numbers for which the radicand is greater than or equal to 0 The domain of any radical function with odd index is 0000 Example 7 State the domain of the function Write your answer using interval notation x 3 fx x7 Example 8 State the domain of the function Write your answer using interval notation Z x 5x 4 x g x2 16 Example 9 State the domain of the function Write your answer using interval notation hx W Example 10 State the domain of the function Write your answer using interval notation hx 3xx2 9 Example 11 State the domain of the function Write your answer using interval notation x 1 x S fx Here is a method for nding the range of a function If f x y solve y for x and use the methods for nding domain to state the range of the function x2 Example 12 Find the domain and range of the function gx 5 x The easiest way to nd the range of a function is to look at the graph of the function We ll revisit nding the range after we do some graphing You also need to be able to evaluate a function at a given value of x or at an expression Example 13 If fx x2 3x 1 find f0 f l f5 ft ft3 and ft 3 2x 3 x lt 1 Example 14 If fx x2 2x 3 l S x S 3 find f3 f0 f5 and f 3 6x x gt3 Next you will need to be able to form a difference quotient To nd a difference quotient you will compute W assuming that h 72 0 You can do this in three steps Compute fx h Then compute fx h fx fx h fx h Then compute Example 15 Find the difference quotient f x 4x 7 Example 16 Find the difference quotient fx x2 2x 9 Functions and Graphs The graph of a function is the set of ordered pairs x y where x is in the domain of fx andy fx You should be able to tell instantly if a graph is the graph of a function using the Vertical Line Test V LT According to the VLT a graph is the graph of a function f x if any vertical line which you can draw will intersect the graph of f x in at most one point Example 17 Determine which of these graphs are graphs of functions A B s You should remember how to graph using a table of values and plotting points This is a very ineff1cient way to graph so we won t do it much during this semester Much of Math 1330 has to do with learning quick and accurate ways to graph functions Point plotting should be used as a last resort With that said here is an example Example 18 Make a table of values and sketch fx Ix 4 Sometimes you will need to graph a piecewise de ned function Here s an example 2 x xgt2 Example 19 Graph fx x2 3SxSZ 3 xlt 3 7 You should be able to state the domain and range of a function given its graph Example 20 State the domain and the range of the function that is graphed You should be able to find the x and y intercepts of the graph of a function given the function Example 21 Find the x and y intercepts of the graph of f x 4x 8 Example 22 Find the x and y intercepts of the graph of f x x2 4 You should be familiar With some vocabulary from College Algebra You should know What we mean by a maximum value a minimum value a turning point and an increasing function and a decreasing function A maximum value is the biggest y value a function takes on A function may or may not have a maximum value Example 23 State the maximum value of the function that is graphed a Example 24 State the maximum value of the function that is graphed A minimum value is the smallest y value a function takes on A function may or may not have a minimum value Example 25 State the minimum value of the function that is graphed 6 Example 26 State the minimum value of the function that is graphed Example 27 State the minimum value of the function that is graphed A function is increasing on an interval ab if f x1 lt f x2for each x1 lt x2 in a b You can think a function is increasing if the y values are getting bigger as we look from left to right A function is decreasing on an interval a b if fx1 gt fx2for eachx1 lt x2 in ab You can think a function is decreasing if the y values are getting smaller as we look from left to right Example 28 State the intervals on which the function graphed is increasing and the intervals on which it is decreasing Example 29 State the intervals on which the function graphed is increasing and the intervals on which it is decreasing A turning point is a point where the graph of a function changes from increasing to decreasing or where it changes from decreasing to increasing Example 30 Identify any turning points on the graph of this function a The next two topics may be new to you We can identify certain functions as even functions or odd functions These functions have certain symmetries and knowing that a function is even or odd is another piece of information to help you graph it efficiently Tests for Symmetry A function has symmetry in the x axis if x y is on the graph of f whenever x y is Test leave x alone and substitute 7y for y If you get an equivalent equation your function has symmetry in the x axis A function has symmetry in the y axis if x y is on the graph of f whenever x y is Test leave y alone and substitute 7x for x If you get an equivalent equation your function has symmetry in the y axis A function has symmetry in the origin if x y is on the graph of f whenever x y is Test substitute 7x for x and 7y for y If you get an equivalent equation your function has symmetry in origin Note an equation can have more than one type of symmetry Example 32 Test the equation for symmetry y2 xy 2 Example 33 Test the equation for symmetry x2 y2 15 A function is even if f x f x for all x in the domain of the function Even functions are symmetric with respect to the y aXis A very common even function is f x x2 whose graph is shown here A function is odd if f x f x for all x in the domain of the function Odd ll lCthl lS have symmetry with respect to the origin Here are a couple of examples of odd functions 1 3 x x f 2 11 Example 34 Determine if fx 5x4 3x2 2 is odd even or neither Example 35 Determine if f x x3 x is odd even or neither Example 36 Determine if f x x3 x 1is odd even or neither Transformations of Graphs In College Algebra you should have learned to transform nine basic functions Here are the basic functions You should know the shapes of each graph domain and range of the function and you should be able to state intervals on which the function is increasing and intervals on which the function is decreasing fxx 7 7 7 4 4 1 W 3 A gt a fxV3x 5 4 1 is I 4 I 1 1 x D is 75 7 73 7 r You should be able to translate these graphs vertically andor horizontally re ect them about the x or the y axis and stretch them or shrink them vertically or horizontally You may nd it helpful to apply transformations in this order Vertical and or Horizontal Stretching or Shrinking Re ection in the x aXis Horizontal or Vertical translations Re ection in the y aXis bP N This is not the only order which works but you will make few mistakes if you apply transformations in this order Before we start with some examples let s take the basic function f x J and apply some transformations to this function fx fxl fx3 fx2 fx1 2fOC 2fx3 3fx 15 f 96 Example 37 Suppose you are asked to graph the function 7 3f2 7 x 4 Starting with the graph of f x state the transformations needed and the order in which you would apply them in order to sketch 7 3f2 7 x 4 Example 38 Sketch fx 7x 7 32 2using transformations Example 39 Sketch fx 3V 4 x 1 using transformations Sometimes you ll be given a graph with no statement of the function and you ll need to be able to graph a transformed version of it In this case it will be helpful to look at key points on the graph of the function transform those and graph them as a guide to graphing the transformed version Here is an example Example 40 Suppose you are given the graph of a function f x Use it to sketch fx 31 fx Now why does the order in which we apply the transformations matter Let s look at the ll lCthl l f x 2 x2 Two transformations are required 1 a re ection about the x aXis and 2 a vertical shift of 2 According to our order for applying transformations we should do them in this order so that the resulting graph will look like this If we apply them in the opposite order the sketch will look like this 7 The first one is correct and the second one is wrong Why Now suppose you are given the graph of a ll lCthl l and you are asked to write the ll lCthl l You ll need to be able to identify the basic ll lCthl l that s given and then describe all of the transformations that were applied to it From that you should be able to write the ll lCthl l Example 41 Write the function that is graphed here Example 42 Write the function that is graphed here V Combining Functions We can combine functions in any of five ways Four of these are the familiar arithmetic operations addition subtraction multiplication and division and are very intuitive The fifth type of combining functions is called composition of functions In all cases we ll be interested in combining the functions and in nding the domain of the combined function Suppose we have two functions f x with domainA and gx with domain B f gx fx gx with domain xlx E A n B f gx fx gx with domain xx e A n 3 f x fxgx with domain xix e A n B g 100 f with domain xix e A n B gx 7t 0 g gx Example 43 Suppose fx 2x 3 and gx 4x 1 Find f g f g fg and Land state the domain of each g The nal way of combining functions is called composition of functions f o gx fgx with domain x xe B gx e A Example 44 Suppose fx 4x 3 and gx 2x 1 Find f o g and g o f and state the domain of each Example 45 Suppose fx x2 4x 3 and gx 3x 1 Find each ofthe following a f g2 b gf1 0 fg0 d gg3 e 5 g f gfx