Chapter 5: Inverse, Exponential, and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions 5.3 Logarithms and Their Properties 5.4 Logarithmic Functions 5.5 Exponential and Logarithmic Equations and Inequalities 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions

One-to-One Functions Is the function a one-to-one function? If so, create the inverse function 𝑓 −1 𝑥 = Numerically (Table): Age (years) Height (inches) 5 40 9 45 13 60 17 66 19

One-to-One Functions Is the function a one-to-one function? If so, create the inverse function 𝑓 −1 𝑥 = Numerically (Table): Age (years) Height (inches) 5 40 9 45 13 60 17 66 19 Height (inches) Age (years) 40 5 45 9 60 13 66 17 19

One-to-One Functions Given a word description, what would the inverse be: You go into class, sit down, get your book out of your backpack, and open your book.

One-to-One Functions Given a word description, what would the inverse be: You go into class, sit down, get your book out of your backpack, and open your book. Inverse: You close your book, put your book in your backpack, get up and exit the classroom.

Horizontal Line Test If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one. Example Use the horizontal line test to determine whether the graphs are graphs of one-to-one functions.

Horizontal Line Test If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one. Example Use the horizontal line test to determine whether the graphs are graphs of one-to-one functions. No Yes

Graphing the inverse relation or function Draw the inverse relation, then the inverse function. No Yes

One-to-One Functions

Inverse Functions Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if

Inverse Functions Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if

Finding an Equation for the Inverse Function Verbally and Algebraically Finding the Equation of the Inverse of y = f(x) For a one-to-one function f defined by an equation y = f(x), find the defining equation of the inverse as follows. (Any restrictions on x and y should be considered.)

Example of Finding f -1(x)

Example of Finding f -1(x)

Example of Finding f -1(x) 1. h(x) = 5 𝑥 + 9   2. f(x) = 4x3 - 5

Finding the Inverse of a Function with a Restricted Domain

Finding the Inverse of a Function with a Restricted Domain Solution Notice that the domain of f is restricted to [–5, ), and its range is [0, ). It is one-to-one and thus has an inverse.

Application of Inverse Functions Example Use the one-to-one function f(x) = 3x + 1 and the numerical values in the table to code the message PLAY MY MUSIC. A 1 B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 K 11 L 12 M 13 N 14 O 15 P 16 Q 17 R 18 S 19 T 20 U 21 V 22 W 23 X 24 Y 25 Z 26

Application of Inverse Functions Example Use the one-to-one function f(x) = 3x + 1 and the numerical values in the table to code the message PLAY MY MUSIC. A 1 B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 K 11 L 12 M 13 N 14 O 15 P 16 Q 17 R 18 S 19 T 20 U 21 V 22 W 23 X 24 Y 25 Z 26