6.2Inverse Functions

A function f is one-to-one if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain.

A function is not one-to-one if two different elements in the domain correspond to the same element in the range.

x1

x2

x3

y1

y2

y3

x1

x3

y1

y2

y3

x1

x2

x3

y1

y3Domain

Domain

Domain

Range

Range

RangeOne-to-onefunction

Not a function

NOT One-to-onefunction

Theorem Horizontal Line Test

If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.

Use the graph to determine whether the function f x x x( ) 2 5 12

is one-to-one.

Not one-to-one.

Use the graph to determine whether the function is one-to-one.

One-to-one.

.Let f denote a one-to-one function y = f(x). The inverse of f, denoted by f -1 , is a function such that f -1(f( x )) = x for every x in the domain of f and f(f -1(x))=x for every x in the domain of f -1.

Input x f(x) f -1(f( x ))=xApply f Apply f -1

Input x f -1(x) f(f -1(x))=xApply f -1 Apply f

Domain of f Range of f

Range of f 1 Domain of f 1

f 1

f

Domain of Range of

Range of Domain of

f f

f f

1

1

Theorem

The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

f 1

2 0 2 4 6

2

2

4

6 f

f 1

y = x

(2, 0)

(0, 2)

Procedure for Finding the Inverse of a One-to-One Function

• In y = f(x) interchange the variables x and y to obtain x = f(y)

• If possible, solve the implicit equation for y in terms of x to obtain the explicit form of f -1.

• Check the results by showing that f -1(f( x )) = x and f(f -1(x)) =

x

y= f -1(x)

Find the inverse of

The function is one-to-one.Interchange variables.

Solve for y.

Check.