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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.

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Presentation Transcript
slide2

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.

slide3

x1

y1

x1

y1

x2

y2

x2

x3

x3

y3

y3

Domain

Range

Domain

Range

One-to-one

function

NOT One-to-one

function

x1

y1

y2

x3

y3

Not a

function

Domain

Range

slide4

M:Mother Function is NOT one-one

Joe

Samantha

Anna

Ian

Chelsea

George

Laura

Julie

Hilary

Barbara

Sue

Humans

Mothers

slide5

S: Social Security function IS one-one

Joe

Samantha

Anna

Ian

Chelsea

George

123456789

223456789

333456789

433456789

533456789

633456789

Americans

SSN

slide8

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.

slide11

The inverse of the social security function

Joe

Samantha

Anna

Ian

Chelsea

George

123456789

223456789

333456789

433456789

533456789

633456789

SSN

Americans

slide14

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

.

slide15

Domain of f

Range of f

slide16

Theorem

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

slide17

y = x

(0, 2)

(2, 0)

finding the inverse of a 1 1 function
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.

find the inverse of also find its domain and range
Find the inverse of Also find its domain and range

Step1:

Step2: Interchange x and y

Step 3: Solve for y