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One-to-One Functions; Inverse Function

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Page 1: 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

One-to-One Functions; Inverse Function

Page 2: 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

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.

Page 3: 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

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

Page 4: 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

M: Mother Function is NOT one-one

Joe

Samantha

Anna

Ian

Chelsea

George

Laura

Julie

Hilary

Barbara

Sue

Humans Mothers

Page 5: 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

S: Social Security function IS one-one

Joe

Samantha

Anna

Ian

Chelsea

George

123456789

223456789

333456789

433456789

533456789

633456789

Americans SSN

Page 6: 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

Is the function f below one – one?

1

2

3

4

5

6

7

10

11

12

13

14

15

16

Page 7: 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

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.

Page 8: 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

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

is one-to-one.

Not one-to-one.

Page 9: 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

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

One-to-one.

Page 10: 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

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

and y

Page 11: 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

123456789

223456789

333456789

433456789

533456789

633456789

SSN

Joe

Samantha

Anna

Ian

Chelsea

George

Americans

The inverse of the social security function1S

Page 12: 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

Let and

Find

3)( xxf 31

)( xxg

))(())(( xfgandxgf

Page 13: 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

g is the inverse of f.

311 )( xxf

Page 14: 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

.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.

Page 15: 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

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

Page 16: 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

Theorem

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

f 1

Page 17: 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

2 0 2 4 6

2

2

4

6 f

f 1

y = x

(2, 0)

(0, 2)

Page 18: 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

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

Page 19: 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

Find the inverse of

Step1:

Step2: Interchange x and y

Step 3: Solve for y

3

5

xy

3

5)(

xxf

3

5

yx

Page 20: 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

x

xxf

35)(1

Page 21: 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
Page 22: 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

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