Inverse Functions Algebra III, Sec. 1.9 Objective You will learn how to find inverse functions graphically and algebraically.

Important Vocabulary Inverse Function - a function that reverses another function Horizontal Line Test – a method to determine if a function is one-to-one

Inverse Functions For a function f that is defined by a set of ordered pairs, to form the inverse function of f, interchange the first and second coordinates of each of the ordered pairs. For a function f and its inverse f-1, the domain of f is equal to the range of f-1, and the range of f is equal to the domain of f-1.

Inverse Functions To verify that two functions, f and g, are inverse functions of each other, perform the composition of each function.

Example Find the inverse of f(x) = 8x. Then verify that both f(f- 1(x)) and f-1(f(x)) are equal to the identity function. The function f multiplies each input (x) by 8. To “undo” this function, you need to divide each input by 8. So…

Example 1 Find the inverse of f(x) = 8x. Then verify that both f(f- 1(x)) and f-1(f(x)) are equal to the identity function.

Example 2 Which of these functions is the inverse of f(x) = 5x + 8? or ✔

Example 2 (cont.) Which of these functions is the inverse of f(x) = 5x + 8? or ✖

Example (on your handout) Verify that the functions and are inverse functions of each other. ✔

The Graph of an Inverse Function If the point (a, b) lies on the graph of f, then the point (b, a) must lie on the graph of f-1 and vice versa. The graph of f-1 is a reflection of the graph of f in the line y = x.

Example 3 Sketch the graphs of the inverse functions on the same rectangular coordinate system, and show that the graphs are reflections of each other in the line y = x. x f-1(x) x f-1(x) -7 -2 -4 -1 2 1 5 x f(x) -2 -1 1 2 x f(x) x f(x) -2 -7 -1 -4 1 2 5

Example 4 Sketch the graphs of the inverse functions on the same rectangular coordinate system, and show that the graphs are reflections of each other in the line y = x.

One-to-One Functions To tell whether a function has an inverse function from its graph, use the horizontal line test. A function f is one-to-one if each value of the dependent variable corresponds to exactly one value of the independent variable. A function f has inverse function if and only if f is one-to-one. * No horizontal line intersects the graph at more than one point * It passes the horizontal line test

Example 5 Use the graph of the function and the Horizontal Line Test to determine whether the function has an inverse. Does it pass the horizontal line test? Yes So f has an inverse

Example (on your handout) Does the graph of the function have an inverse function? Explain. Does it pass the horizontal line test? No So f does not have an inverse

Finding Inverse Functions Algebraically To find the inverse of a function f algebraically, Use the horizontal line test to decide whether f has an inverse function. In the equation for f(x), replace f(x) with y. Interchange the roles of x and y, and solve for y. Replace y with f-1(x) in the new equation. Verify that f and f-1 are inverse functions of each other by showing that the domain of f is the range of f-1 and vice versa, and the compositions give you the identity x.

Example 6 Find the inverse of f(x) = -4x – 9.

Example 7 Find the inverse of

Example (on your handout) Find the inverse of f(x) = 4x – 5.

Practice Sec 1.9, pg 90 – 91 # 23, 27, 35 (also sketch f and f-1), 37-40, 57, 69, 71