1.7: Inverse Relations & Functions Pre-Calculus Unit 1

Objectives Use the graphs of functions to determine if they have inverse functions. Find inverse functions algebraically and graphically.

Inverse Relations Exist if and only if one relation contains (b, a) whenever the other contains (a, b) In equation form, the inverse relation can be found by interchanging the independent and dependent variables

Inverse Function The inverse relation of a function f is also a function 𝑓 −1 f inverse Not all functions have inverse functions

Example 1 A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no.

Example 1 B. Graph the function f (x) = x 5 + x 3 – 1 using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no.

One-to-One A function that passes the horizontal line test A one-to-one function f has an inverse 𝑓 −1 such that 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

Exit Slip Determine whether f has an inverse function for . If it does, find the inverse function and state any restrictions on its domain. A. B. C. D. f –1(x) does not exist

Example 2 A. Determine whether f has an inverse function for . If it does, find the inverse function and state any restrictions on its domain.

Example 2 B. Determine whether f has an inverse function for . If it does, find the inverse function and state any restrictions on its domain.

1.7 Day 1 HW

An inverse function 𝑓 −1 has the effect of “undoing” the action of a function f

Example 3A

Example 3B Show that f (x) = x 2 – 2, x  0 and are inverses of each other.

1.7 Day 2 HW 3 4

5 6 7

Example 4 A. Use the graph of relation A to sketch the graph of its inverse.

Example 4 B. Use the graph of the function to graph its inverse function.

Example 5 A. The fixed costs for manufacturing one type of stereo system are $96,000 with variable cost of $80 per unit. The total cost f (x) of making x stereos is given by f(x) = 96,000 + 80x. Explain why the inverse function f –1(x) exists. Then find f –1(x).

Example 5 B. What do f –1(x) and x represent in the inverse function?

Example 5 C. What restrictions, if any, should be placed on the domain of f (x) and f –1(x)? Explain.

Example 5 D. Find the number of stereos made if the total cost was $216,000.