One-to-One Functions; Inverse Function. A function f is one-to-one if for each x in the domain of f...

One-to-One Functions; Inverse Function

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

M: Mother Function is NOT one-one

Joe

Samantha

Anna

Ian

Chelsea

George

Laura

Julie

Hilary

Barbara

Sue

Humans Mothers

S: Social Security function IS one-one

Joe

Samantha

Anna

Ian

Chelsea

George

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633456789

Americans SSN

Is the function f below one – one?

1

2

3

4

5

6

7

10

11

12

13

14

15

16

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.

The inverse of a one-one function is obtained by switching the role of x

and y

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223456789

333456789

433456789

533456789

633456789

SSN

Joe

Samantha

Anna

Ian

Chelsea

George

Americans

The inverse of the social security function1S

Let and

Find

3)( xxf 31

)( xxg

))(())(( xfgandxgf

g is the inverse of f.

311 )( xxf

.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.

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)

Finding the inverse of a 1-1 function

Step1: Write the equation in the form

Step2: Interchange x and y.

Step 3: Solve for y.

Step 4: Write for y.

)(xfy

)(1 xf

Find the inverse of

Step1:

Step2: Interchange x and y

Step 3: Solve for y

3

5

xy

3

5)(

xxf

3

5

yx

x

xxf

35)(1