Functions - Linda Green

FUNCTIONSFunctionsAfter completing this section, students should be able to: Decide whether a relationship between input and output values is a function ornot, based on an equation, a graph, or a table of values. Find the corresponding output value for a given input value for a function givenin equation, graphical, or tabular form. Find the corresponding input value(s) for a given output value for a function givenin equation, graphical, or tabular form. Find the domain and range of a function or relation based on a graph or table ofvalues. Find the domains of functions given in equation form involving square roots anddenominators. .2

FUNCTIONSDefinition. A function is correspondence between input numbers (x-values) and output numbers (y-value) that sends each input number (x-value) to exactly one outputnumber (y-value).Sometimes, a function is described with an equation.Example. y x2 1, which can also be written as f (x) x2 1What is f (2)?f (5)?What is f (a 3)?3

FUNCTIONSSometimes, a function is described with a graph.Example. The graph of y g(x) is shown belowWhat is g(2)?g(5)?4

FUNCTIONSDefinition. The domain of a function is all possible x-values. The range is the y-values.Example. What is the domain and range of the function g(x) graphed below?5

FUNCTIONSExample. What are the domains of these functions?xA. g(x) 2x 4x 3B. f (x) p32x6

FUNCTIONSC. h(x) px232x4x 3END OF VIDEO7

FUNCTIONSDefinition. A relation is .Example. Which of these relations represent functions?1. {(4, 4), (6, 4), ( 4, 6), (7, 4)}2.13. y x224. y2 3x5.8

FUNCTIONSExample. Find the domain of h(x) 2x9p 6x 59

FUNCTIONSp2 xExample. Find the domain of g(x) x 3A. ( 1, 2)B. ( 1, 2]C. ( 3, 2]D. ( 1, 3) [ ( 3, 2)E. ( 1, 3) [ ( 3, 2]10

INCREASING AND DECREASING FUNCTIONS, MAXIMUMS AND MINIMUMSIncreasing and Decreasing Functions, Maximums and MinimumsAfter completing this sections, students should be able to: Identify the intervals on which a function is increasing and decreasing based on agraph. Define absolute maximum and minimum values and points and local minimumvalues and points. Identify absolute and local max and min values and points based on a graph.11

INCREASING AND DECREASING FUNCTIONS, MAXIMUMS AND MINIMUMSExample. Which function is increasing? Which is decreasing?.Example. On what intervals is the function graphed below increasing? Decreasing?12

INCREASING AND DECREASING FUNCTIONS, MAXIMUMS AND MINIMUMSDefinition. A function f (x) has an absolute maximum at x c ifThe y-value f (c) is called theand the point (c, f (c)) is calledDefinition. A function f (x) has an absolute minimum at x c ifThe y-value f (c) is called theand the point (c, f (c)) is calledDefinition. Absolute maximum and minimum values can also be called13

INCREASING AND DECREASING FUNCTIONS, MAXIMUMS AND MINIMUMSDefinition. A function f (x) has an local maximum at x c ifThe y-value f (c) is called theand the point (c, f (c)) is calledDefinition. A function f (x) has an local minimum at x c ifThe y-value f (c) is called theand the point (c, f (c)) is calledDefinition. Local maximum and minimum values can also be called14

INCREASING AND DECREASING FUNCTIONS, MAXIMUMS AND MINIMUMSExample. .1. Mark all local maximum and minimum points.2. Mark all absolut maximum and and minimum points.3. What are the local maximum and minimum values of the function?4. What are the absolute maximum and minimum values of the function?END OF VIDEOS15

INCREASING AND DECREASING FUNCTIONS, MAXIMUMS AND MINIMUMSQuestion. What is the di erence between a maximum point and a maximum value?Question. What is di erence between an absolute maximum value and a local maximumvalue?Question. Is it possible to have more than one absolute maximum value? More thanone absolute maximum point?16

INCREASING AND DECREASING FUNCTIONS, MAXIMUMS AND MINIMUMSExample. Find the local maximum points, the local minimum points, the absolutemaximum points, and the absolute minimum points.What are the absolute maximum and minimum values?17

INCREASING AND DECREASING FUNCTIONS, MAXIMUMS AND MINIMUMSExample. On what intervals is the function graphed below increasing? Decreasing?What are its absolute max and min points? Absolute max and min values?18

SYMMETRY AND EVEN AND ODD FUNCTIONSSymmetry and Even and Odd FunctionsAfter completing this sections, students should be able to: Identify whether a graph is symmetric with respect to the x-axis, symmetric withrespect to the y-axis, symmetric with respect to the origin, or none of these. Determine whether a function is even or odd or neither, based in its graph or itsequation. Explain the relationship between even and odd functions and the symmetry oftheir graphs.19

SYMMETRY AND EVEN AND ODD FUNCTIONSDefinition. A graph is symDefinition. A graph is sym- Definition. A graph is symmetric with respect to the orimetric with respect to the x- metric with respect to the ygin if .axis if .axis if .Whenever a point (x, y) is on Whenever a point (x, y) is on Whenever a point (x, y) is onthe graph, the pointthe graph, the pointthe graph, the pointis also on the graph.is also on the graph.20is also on the graph.

SYMMETRY AND EVEN AND ODD FUNCTIONSExample. Which graphs are symmetric with respect to the x-axis, the y-axis, the origin,or neither?21

SYMMETRY AND EVEN AND ODD FUNCTIONSDefinition. A function f (x) is even if .Example. f (x) x2 3 is even because .Definition. A function f (x) is odd if .Example. f (x) 5x1is odd because .x22

SYMMETRY AND EVEN AND ODD FUNCTIONSQuestion. There is no word like even or odd for when a function’s graph is symmetricwith respect to the x-axis. Why not?END OF VIDEO23

SYMMETRY AND EVEN AND ODD FUNCTIONSExample. Are these graphs symmetric with respect to the x-axis, the y-axis, the origin,or neither?2y xy x33xy2 xy3 x2y2 x 24

SYMMETRY AND EVEN AND ODD FUNCTIONSExample. Do these equations have graphs that are symmetric with respect to the x-axis,the y-axis, the origin, or neither?2y 3 xx2 2y4 6y x xy x x2xWhich equations represent even functions? Odd functions?25

SYMMETRY AND EVEN AND ODD FUNCTIONSExample. Determine whether the functions are even, odd, or neither.1. f (x) 4x3 2x2. g(x) 5x43x2 13. h(x) 2x3 7x226

TRANSFORMING FUNCTIONSTransforming FunctionsAfter completing this section, students should be able to Identify the motions corresponding to adding or multiplying numbers or introducing a negative sign on the inside or the outside of a function. Draw the transformed graph, given an original graph of y f (x) and an equationlike y 3 f (x 2), using a point by point analysis or a wholistic approach. Identify the equation for transformed graphs of toolkit functions like y x andy x2 Identify a point on a transformed graph, given a point on the original graph andthe equation of the transformed graph.27

TRANSFORMING FUNCTIONSReview of Function NotationpExample. Rewrite the following, if g(x) x.a) g(x)2 b) g(x2) c) g(3x) d) 3g(x) e) g( x) pExample. Rewrite the following in terms of g(x), if g(x) x.pf) x 17 pg) x 12 ph) 36 · x qi) 14 x 28

TRANSFORMING FUNCTIONSExample. Graphp y xp y x 2p y x 229

TRANSFORMING FUNCTIONSRules for transformations: Numbers on the outside of the function a ect the y-values and result in verticalmotions. These motions are in the direction you expect. Numbers on the inside of the function a ect the x-values and result in horizontalmotions. These motions go in the opposite direction from what you expect. Adding results in a shift (translations) Multiplying results in a stretch or shrink A negative sign results in a reflection30

TRANSFORMING FUNCTIONSExample. Consider g(x) pto the graph of y x?pa) y x 4b) y px 12c) y 3 ·d) y px. How do the graphs of the following functions compareqpx14xEND OF VIDEO31

TRANSFORMING FUNCTIONSRules of Function Transformations (see graph animations involving y sin(x)) A number added on the OUTSIDE of a function . A number added on the INSIDE of a function . A number multiplied on the OUTSIDE of a function . A number mulitplied on the INSIDE of a function . A negative sign on the OUTSIDE of a function . A negative sign on the INSIDE of a function .32

TRANSFORMING FUNCTIONSExample. Consider h(x) x3. How do the graphs of the following functions compareto the graph of y x3?a) y (x 1)3b) y (2x)3x3c) y 25d) y ( x)3 333

TRANSFORMING FUNCTIONSNote. There are two approaches to graphing transformed functions:34

TRANSFORMING FUNCTIONSExample. The graph of a certain function y f (x) is shown below.Use transformations to draw the graph of the function y your final graph.35f (2x) 4. Label points on

TRANSFORMING FUNCTIONSExample. The graph of a certain function y f (x) is shown below.Use transformations to draw the graph of the function y 3 f ( x)your final graph.361. Label points on

TRANSFORMING FUNCTIONSExample. Given the original graph if y f (x)Find the graph of y 3 f ( x) 1, and write down the equations of the other graphs.37

TRANSFORMING FUNCTIONSExtra Example. Suppose the graph of y f (x) contains the point (3, 1). Identify apoint that must be on the graph of y 2 f (x 1).A. (2, 1)B. (2, 1)C. (4, 1)D. (4, 2)38

PIECEWISE FUNCTIONSPiecewise FunctionsAfter completing this sections, students should be able to: Evaluate piecewise functions at a given x-value. Graph piecewise functions.39

PIECEWISE FUNCTIONSExample. The function f is defined as follows:8 2 if x 1 xf (x) : 2x 3 if x 11. What is f ( 2)? What is f (1) What is f (3)?2. Graph y f (x).3. Is f (x) continuous?END OF VIDEO40

PIECEWISE FUNCTIONSExample. The function g(x) is defined by:8 (x 2)2 1g(x) x 1 2 :3if 4 x 2if 2 x 2if 2 x 41. What is g( 2)? What is g(1) What is g(3.5)?2. Graph y g(x).3. What are the domain and range of g(x)?41

PIECEWISE FUNCTIONSExtra Example. The function h(x) is defined by:8 2 x ph(x) x :1 x1if 2 x 1if 1 x 4if 4 x 61. What is h(1)? What is h(5)?2. Graph y h(x).3. What are the domain and range of h(x)?42

INVERSE FUNCTIONSInverse functionsAfter completing this section, students should be able to: Based on the graph of a function, determine if the function has an inverse that is afunction. Draw the graph of an inverse function, given the graph of the original. Use a table of values for a function to write a table of values for its inverse. Determine if two given functions are inverses of each other by computing theircompositions. Use a formula for a function to find a formula for its inverse. Find the range of the inverse function from the domain of the original function. Find the domain of the inverse function from the range of the original function.43

INVERSE FUNCTIONSExample. Suppose f (x) is the function defined by the chart below:x 2 3 4 5f (x) 3 5 6 1In other words, f (2) 3 f (3) 5 f (4) 6 f (5) 1Definition. The inverse function for f , written f 1(x), undoes what f does. f 1( 3 ) 2 f 1() f 1() f 1() x3f 1(x) 2Key Fact 1. Inverse functions reverse the roles of y and x.44

INVERSE FUNCTIONSGraph y f (x) and y f 1(x) on the same axes below. What do you notice about thepoints on the graph of y f (x) and the points on the graph of y f 1?Key Fact 2. The graph of y f 1(x) is obtained from the graph of y f (x) by reflectingover the line.45

INVERSE FUNCTIONSIn our same example, compute:ffff1111f (2) f (3) f (4) f (5) Key Fact 3. f 1 f (x) way of saying that f and fffffffffand f f 1(x) 1undo each other.1(3) 1(5) 1(6) 1(1) . This is the mathematicalExample. f (x) x3. Guess what the inverse of f should be. Remember, fthe work that f does.461undoes

INVERSE FUNCTIONSExample. Find the inverse of the function:f (x) 5x3xNote. f 1(x) means the inverse function for f (x). Note that f 1(x) ,471f (x) .

INVERSE FUNCTIONSQuestion. Do all functions have inverse functions? That is, for any function that youmight encounter, is there always a function that is its inverse?Try to find an example of a function that does not have an inverse function.48