Section 8.2Inverse Functions

• Consider the following two tables

• Are they both functions?• What are f-1 and g-1?

x f(x)

-2 10

-1 12

0 16

1 20

2 27

3 44

x g(x)

-5 -10

-2 -12

-1 -16

1 -16

2 -12

5 -10

Definition of an Inverse Function• Suppose Q = f(t) is a function with the

property that each value of Q determines exactly one value of t. The f has an inverse function, f -1 and

If a function has an inverse, it is said to be invertible

).(ifonlyandif)(1 tfQtQf

Graphs of Inverses• Consider the function 3)( xxf

• The inverse is 3)( 21 xxf

Plot of the two graphs together

Horizontal Line Test

• A function must be one-to-one in order to have an inverse (that is a function)

• A function is one-to-one if it passes the horizontal line test– A horizontal line may hit a graph in at most one

point

• We can restrict the domain of functions so their inverse exists– For example, if x ≥ 0, then we have an inverse for

3)( 2 xxg

• Let’s talk about the following problems

)100(find1)(Given

)20(find23)(Given

)(find13)(Given

)(find13)(Given

15

13

1

13

kxxxk

hxxh

xgxg

xfxxf

x

x

Property of Inverse Functions• If y = f(x) is an invertible function and y = f -1(x) is

its inverse, then

definedis)(

whichforofvaluesallfor))((

definedis)(

whichforofvaluesallfor))((

1

1

1

xf

xxxff

xf

xxxff

• In your groups try problems 5, 15, and 25