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# Chapter 3: Functions and Their Graphs MAT 109

Barry University

GPA 3.7

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This 22 page Bundle was uploaded by Sterling Notetaker on Wednesday September 7, 2016. The Bundle belongs to MAT 109 at Barry University taught by Dr. Singh in Fall 2016. Since its upload, it has received 9 views. For similar materials see Precalculus Mathematics 1 in Mathmatics at Barry University.

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Date Created: 09/07/16

MAT 109 PreCalculus Mathematics 1 3.1 Functions Notes L. Sterling August 22nd, 2016 Abstract Give a de▯nition to each of the terms listed in this section. 1 Relation A correspondence between two sets 2 Input/Abscissa The relation of x or the x-values 3 Output/Ordinate The relation of y or the y-values 4 Map Helps illustrate a relation by using a set of inputs and drawing arrows to the corresponding element in the set of outputs. 5 Ordered pairs Can be used to represent as (x;y). 6 Functions Relations between a set of inputs (x-values) and a set of outputs (y-values) 1 7 Domain The set of all possible input values 8 Range The set of all possible output values 9 Independent Variable The variable x since it can be assigned any number (s) from the given domain. 10 Dependent Variable The variable y since its own value is depending on x. 11 Argument A function’s independent variable. 12 Di▯erence Quotient f(x+h)▯f(x) h is one of the important expressions in calculus. 13 Implicitly/Implicit Form When a function has to be solved for y to make it explicit. 14 Explicitly/Explicit Form When a function is in the form of either y = mx + b, y = mx, or y = b. 15 Is De▯ned A point, or f(x), that does exist when it comes to ▯nding the domain of a function. 16 DNE/Does Not Exist When a point, or f(x), does not exist in the domain of a function, or f. 2 17 Sum of Two Functions (f + g)(x) = f(x) + g(x) 18 Di▯erence of Two Functions (f ▯ g)(x) = f(x) ▯ g(x) 19 Product of Two Functions (f ▯ g)(x) = f(x) ▯ g(x) = f(x)g(x) 20 Quotient of Two Functions f f(x) (f=g)(x) =g()(x) =g(x)g(x)6=0 3 MAT 109 PreCalculus Mathematics 1 3.2 The Graph of a Function Notes L. Sterling August 24th, 2016 Abstract Give a de▯nition to each of the terms listed in this section. 1 Graphs The functional set of points (x;y) on an xy-plane to note the given equation. 2 VLT: Vertical Line Test A general set of points that are located on a xy-plane is a function’s veri▯ed graph i▯ (if and only if) every point intersects the vertical line of a graph only once. 3 Things to Obtain about a Graph Say you have a function; the following parts are some of the things you would need to obtain from and/or about the graph of any function: 3.1 Domain Domains are general sets of all of the x-coordinates, which are the ▯rst half of the elements of any ordered pair. 3.2 Range Ranges are general sets of all of the y-coordinates, which are the second half of the elements of any ordered pair. 1 3.3 If a point is on the graph or not This is basically saying asking the point (or points) that you were given are solution(s) to the given equation. If that case, then all you have to do is "plug and play". Here is an example: f(x) = 5x + 7 f(x) = 5x + 7 Point : (0;8) Point : (1;12) f(x) = 5x + 7 f(x) = 5x + 7 8 = 5(0) + 7 12 = 5(1) + 7 8 = 0 + 7 12 = 5 + 7 8 = 7 12 = 12 8 6= 7 12 = 12 3.4 Solve for y when given x 2 f(x) = 5x + 7 Point : (5;f(x)) 2 f(x) = 5x + 7 2 f(x) = 5(5) + 7 f(x) = 5(25) + 7 f(x) = 125 + 7 f(x) = 132 Point : (5;132) 3.5 Solve for x when given y To do so, plug in y and play around to ▯nd x. Here is an example: f(x) = 5x + 7 Point : (x;37) f(x) = 5x + 7 37 = 5x + 7 37 ▯ 7 = 5x + 7 ▯ 7 30 = 5x 30 5x 5 = 5 x = 6 Point : (6;37) 2 3.6 x-intercepts (if any) To do so, let y = 0 and play around to ▯nd x. Here is an example: f(x) = x ▯ 9 Point : (x;0) 2 f(x) = x ▯ 9 0 = x ▯ 9 0 + 9 = x ▯ 9 + 9 9 = x2 x = 9 p p x = 9 x = ▯3 Point : (▯3;0);(3;0) 3.7 y-intercepts (if any) To do so, let x = 0 and play around to ▯nd y. Here is an example: f(x) = x ▯ 9 Point : (0;f(x)) f(x) = x ▯ 9 f(x) = (0) ▯ 9 f(x) = 0 ▯ 9 f(x) = ▯9 Point : (▯9;0) 3.8 Listing points on the graph Look left and right for the x ▯ points: Look up and down for the y ▯ points: 3 MAT 109 PreCalculus Mathematics 1 3.3 Properties of Functions Notes L. Sterling August 24th, 2016 Abstract Give a de▯nition to each of the terms listed in this section. 1 Even Functions When every x-value in its given domain is also in their respected ▯x-values: f(▯x) = f(x) and (x;y) ! (▯x;y) 2 Odd Functions When every x-value in its given domain is also in their respected ▯x-values: f(▯x) = ▯f(x) and (x;y) ! (▯x;▯y) 3 Ways to tell Even and Odd functions 3.1 Even Functions I▯ (if and only if) the given graph is actually symmetric, but with respect to the y-axis. 3.2 Odd Functions I▯ (if and only if) the given graph is actually symmetric, but with respect to the origin. 4 When functions are Increasing, Decreasing, or Constant from its own Graph 4.1 Increasing When it’s on an open interval, I, if, for any possible 1hoice 2f both x and x in I when noting th1t x2< x , which therefore not1s that2f(x ) < f(x ). 1 4.2 Decreasing When it’s on an open interval, I, if, for any possible ch1ice o2 both x and x in I when noting that x < x , which therefore notes that f(x ) > f(x ). 1 2 1 2 4.3 Constant When it’s on an open interval, I, if, for any possible choice of just x in I when the values in f(x) are all equal. 5 Local Maximum A function has one at v if there’s an I, an open interval, contains c so that, for all x-values in an I to make it f(x) ▯ f(c). 5.1 Local Maximum’s f(c) Value Local maximum value 6 Local Minimum A function has one at v if there’s an I, an open interval, contains c so that, for all x-values in an I to make it f(x) ▯ f(c). 6.1 Local Minimum’s f(c) Value Local minimum value 7 Absolute Maximum When f notes a function when some interval and that there’s a number in an interval for which f(x) ▯ f(u), which would make f(u). 7.1 Absolute Maximum’s f(c) Value Absolute maximum value 8 Absolute Minimum When a number in an interval for which f(x) ▯ f(u) for all values of x in an interval, which would make f(u). 8.1 Absolute Minimum’s f(c) Value Absolute minimum value 2 9 Extreme Value Theorem If f’s a continuous function with a domain being in a closed interval of [a;b], then f’s going to have both an absolute maximum and an absolute minimum on [a;b]. 10 Average Rate of Change When a and b df(b)▯f(a)l each▯yth▯y and are in a function’s domain can be de▯ned as vr= b▯a ;a 6= b ▯x= ▯x. 11 Secant Line’s Slope When a function’s average rate of change, from both a and b being equaled to the secant line’s slope that contains both points of (a;f(a)) and (b;f(b)) on the graph. 12 How to ▯nd a Secant Line’s Slope 12.1 Finding the average rate of change from the function from two points. 12.2 ▯nding the equation of the secant line that contains (a;f(a)) and (b;f(b)). 12.3 Graph the two together on the same xy-plane. 3 MAT 109 PreCalculus Mathematics 1 3.4 Library of Functions: Piecewise-De▯ned Functions Notes L. Sterling August 25th, 2016 Abstract Give a de▯nition to each of the terms listed in this section. p 1 Fun Facts About f(x) = x 1.1 Domain Set of nonnegative real numbers. 1.2 Range Set of nonnegative real numbers. 1.3 X-intercept 0 therefore the point being (0;0). 1.4 Y-intercept 0 therefore the point being (0;0). 1.5 The Function Itself Neither odd or even. 1.6 Increasing/Decreasing Increasing on the interval of (0;1) and decreasing never. 1.7 Absolute Minimum/Maximum Contains an absolute minimum of 0 at x = 0 with no absolute maximum. 1 p3 2 Fun Facts About f(x) = x 2.1 Domain Set of all real numbers. 2.2 Range Set of all real numbers. 2.3 X-intercept 0 therefore the point being (0;0). 2.4 Y-intercept 0 therefore the point being (0;0). 2.5 The Function Itself This is an odd function since this function is symmetric with respect to the origin. 2.6 Increasing/Decreasing Increasing on the interval of (▯1;1) and decreasing never. 2.7 Absolute Minimum/Maximum Contains no absolute minimum and no absolute maximum. 3 Fun Facts About f(x) = jxj 3.1 Domain Set of all real numbers. 3.2 Range All values of y, but when y is greater than 0, which can be written as fy j y ▯ 0g. 3.3 X-intercept 0 therefore the point being (0;0). 3.4 Y-intercept 0 therefore the point being (0;0). 2 3.5 The Function Itself This is an even function since this function is symmetric with respect to the y-axis. 3.6 Increasing/Decreasing Decreasing on the interval of (▯1;0) and increasing on the interval of (0;1). 3.7 Absolute Minimum/Maximum Contains an absolute minimum of 0 at x = 0 with no absolute maximum. 4 Constant Function f(x) = b 4.1 What is b? b is and must only be a real number. 4.2 Domain All real numbers. 4.3 Range The set that consists of b. 4.4 Graph Horizontal line whose y-intercept is b, which constant functions are all even functions. 5 Identity Function f(x) = x 5.1 Domain All real numbers. 5.2 Range All real numbers. 5.3 Graph Has a slope of one with a y-intercept of 0, which is also an odd function and it increasing over its own domain while bisecting only Quadrants I and III. 3 6 Square Function f(x) = x 6.1 Domain All real numbers. 6.2 Range The set of nonnegative real numbers. 6.3 Graph It’s an even functioned parabola with its intercept at (0;0) where is increases on the interval of (0;1) and decreases on the interval of (▯1;0). 7 Cube Function f(x) =3x 7.1 Domain All real numbers. 7.2 Range All real numbers. 7.3 Graph This is an odd function where it never decreases, but increases on the interval of (▯1;1). 8 Square Root Function f(x) =px 8.1 Domain A set of nonnegative real numbers. 8.2 Range A set of nonnegative real numbers. 8.3 Graph It’s a square root function that’s not an odd or even function whose intercept is the origin and never decreases, but increases on the interval of (▯1;1). 4 9 Cube Root Function p f(x) =3x 9.1 Domain All real numbers. 9.2 Range All real numbers. 9.3 Graph It’s an odd function whose intercept is the origin and never decreases, but in- creases on the interval of (▯1;1). 10 Reciprocal Function f(x) =1 x 10.1 Domain A set of nonzero real numbers. 10.2 Range A set of nonzero real numbers. 10.3 Graph An odd function with no intercepts while it never increases, but decreases on the intervals of (▯1;0) and (0;1). 11 Absolute Value Function f(x) = jxj 11.1 Domain All real numbers. 11.2 Range A set of nonnegative real numbers. 11.3 Graph An even function with its intercept at the origin with itself increasing when the interval is (0;1) and decreasing when the interval is (▯1;0). 5 12 Greatest Integer Function f(x) = int(x) 12.1 Domain All real numbers. 12.2 Range The set of integers. 12.3 Graph A constant piecewise function that is not odd or even, with a y-intercept of 0 and an x-intercept on the interval of [0;1), and a constant pace on the interval in the term of [k;k + 1) which notes k as an integer. 13 Functions Constant f(x) = b Identity f(x) = x Square f(x) = x Cube f(x) = x SquareRoot f(x) =px 3 CubeRoot f(x) = 1 Reciprocal f(x) = x Absolute f(x) = jxj 14 Continuous When a graph with no gaps or holes that can be drawn without lifting a pencil from the paper. 15 Discontinuous When a graph with gaps or holes that can’t be drawn without lifting a pencil from the paper. 16 Step Function A graph of the greatest integer function, f(x) = int(x) is a common example since the graph itself steps up from one value to the next one, which the graph itself looks like a staircase. 6 17 Piecewise-De▯ned Function When functions can be de▯ned by other or di▯erent equations as di▯erent parts of the domain, which one example is the following: ▯ x if x ▯ 0 f(x) = jxj = ▯x if x < 0 7 MAT 109 PreCalculus Mathematics 1 3.5 Graphing Techniques: Transformations Notes 3.6 Mathematical Models: Building Functions Notes L. Sterling August 26th, 2016 Abstract Give a de▯nition to each of the terms listed in this section. 1 Transformations When a function gets either compressed, shifted, or even stretched to become a whole new function. 2 Vertical Shifts 2.1 y=f(x)+k Raising the graph of f by k units by adding k to f(x). y = f(x) + k;k > 0 2.2 y=f(x)-k Lowering the graph of f by k units by subtracting k to f(x). y = f(x) ▯ k;k > 0 3 Horizontal Shifts 3.1 y=f(x+h) Shifting the graph of f to the left by h units by replacing x by x + h. y = f(x + h);h > 0 1 3.2 y=f(x-h) Shifting the graph of f to the right by h units by replacing x by x ▯ h. y = f(x ▯ h);h > 0 4 Compressing 4.1 y=af(x) First by multiplying every y-coordinate of y = f(x) by a, second by stretching the graph of f vertically if and when a > 1, and ▯nally by compressing the graph of f vertically if 0 < a < 1. y = af(x);a > 0 5 Stretching 5.1 y=f(ax) First by replacing x with ax, second by multiplying every x-coordinate of y = f(x) byx, third by stretching the graph of f vertically if and when 0 < a < 1, and ▯nally by compressing the graph of f vertically if a > 1. y = f(ax);a > 0 6 Re ection 6.1 About the X-Axis Re ecting the graph of f about the x-axis by multiplying f(x) by ▯1 to make it the following: y = ▯f(x) 6.2 About the Y-Axis Re ecting the graph of f about the y-axis by multiplying x in f(x) by ▯1 to make the following: y = f(▯x) 7 Mathematical Models: Building Functions To build an analyze functions in real-world problems. 2 7.1 Main Tips (if/when asked) 7.1.1 Express distance or area 7.1.2 Find distance or area 7.1.3 Find domain 7.1.4 Solve for x 7.1.5 Graphing 7.1.6 Looking for the largest or smallest distance or area 7.1.7 Building models 7.1.8 Look for maximum area 3 MAT 109 PreCalculus Mathematics 1 Chapter 3 Study Guide Notes L. Sterling August 27th, 2016 Abstract Give a de▯nition to each of the terms listed in this section. 1 Functions 1.1 Constant Function f(x) = b 1.2 Identity Function f(x) = x 1.3 Square Function f(x) = x 1.4 Cube Function 3 f(x) = x 1.5 Square Root Function f(x) =p x 1.6 Cube Root Function p3 f(x) = x 1.7 Reciprocal Function 1 f(x) =x 1 1.8 Absolute Value Function f(x) = jxj 1.9 Greatest Integer Function f(x) = int(x) 2 De▯nitions 2.1 Absolute Maximum When f notes a function when some interval and that there’s a number in an interval for which f(x) ▯ f(u), which would make f(u). 2.2 Absolute Minimum When a number in an interval for which f(x) ▯ f(u) for all values of x in an interval, which would make f(u). 2.3 Average Rate of Change f(b)▯f(a) Can be found in a function’s domain to be de▯ned avrA= b▯a ;a6=b = ▯x = ▯x while only when both a and b don’t equal each other. 2.4 Constant Function When it’s on an open interval, I, if, for any possible choice of just x in I when the values in f(x) are all equal. 2.5 Decreasing Function When it’s on an open interval, I, if, for any possible choice of both x and x 1 2 in I when noting that1x < 2 , which therefore notes that1f(x ) 2 f(x ). 2.6 Di▯erence Quotient f (x + h) ▯ f (; h6=0 h 2.7 Domain General sets of all of the x-coordinates, which are the ▯rst half of the elements of any ordered pair. 2.8 Even Function When every x-value in its given domain is also in their respected ▯x-values: f(▯x) = f(x) and (x;y) ! (▯x;y). 2 2.9 Function Relations between a set of inputs (x-values) and a set of outputs (y-values) 2.10 Function Notation f(x) y f Function Argument or Independent V ariable x Dependent V ariable y Function of image or x f (x) 2.11 Increasing Function When it’s on an open interval, I, if, for any possible choice o1 both 2 and x in I when noting that x < x , which therefore notes that f(x ) < f(x ). 1 2 1 2 2.12 Local Maximum Functions with one at v if there’s an I, an open interval, contains c so that, for all x-values in an I to make it f(x) ▯ f(c). 2.13 Local Minimum Functions with one at v if there’s an I, an open interval, contains c so that, for all x-values in an I to make it f(x) ▯ f(c). 2.14 Odd Function When every x-value in its given domain is also in their respected ▯x-values: f(▯x) = ▯f(x) and (x;y) ! (▯x;▯y). 2.15 Range General sets of all of the y-coordinates, which are the second half of the elements of any ordered pair. 2.16 Vertical Line Test A general set of points that are located on a xy-plane is a function’s veri▯ed graph i▯ (if and only if) every point intersects the vertical line of a graph only once. 3

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