4-5 INVERSE FUNCTIONS

Objectives:

1. Find the inverse of a function, if it exists.

INVERSE EXAMPLEConversion formulas come in pairs, for

example:

These formulas “undo” each other, so they are inverses.

DEFINITION OF INVERSESTwo functions f and g are inverses if: f(g(x)) = x and g(f(x)) = x

To check if two functions are inverses, perform both compositions and make sure both equal x.

EXAMPLE 1If and

show that f and g are inverses.

YOU TRY!Show that and

are inverses.

INVERSE NOTATIONThe inverse of f is written f -1

f -1(x) is the value of f -1 at x

Note: is not

FINDING INVERSESThe graph of f -1 is the reflection of f

over the line y = xCan be found by

switching x and y in the ordered pairs.

Find equation of f -1

by switching x and yin the equation andsolving for y.

EXAMPLE 2Let f(x) = 4 – x2 for x 0.Sketch the graph of f and f -1 (x) Find a rule for f -1 (x)

YOU TRY!Let g(x) = (x – 4)2 – 1 for x 4.Sketch the graph of g and g -1 (x) Find a rule for g -1 (x)

EXAMPLE 3Suppose a function f has an inverse.

If f(2) = 3, find:

f -1 (3)

f(f -1(3))

f -1(f(2))

YOU TRY!Suppose a function g has an inverse.

If g(5) = 1, find:

g -1 (1)

g -1(g(5))

g(g -1(1))

DO ALL FUNCTIONS HAVE INVERSES?

Reflect the graph of y = x2 over the line y = x.

Is the result a function?

ONE-TO-ONEOnly functions that are one-to-one

have inverses.One-to-one means each x value has

exactly one y value and each y has exactly one x

Can check using horizontal line test.

EXAMPLE 4 Which functions are one-to-one?Which have inverses?

YOU TRY! Is h(x) one-to-one? Does it have an inverse?

EXAMPLE 5State whether each function has an

inverse. If yes, find f -1 (x) and show f(f -1(x)) = f -1(f(x)) = x

YOU TRY!Does have an

inverse? If so, find f -1 (x).