One-to-One Functions; Inverse Function

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# One-to-One Functions; Inverse Function - PowerPoint PPT Presentation

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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## One-to-One Functions; Inverse Function

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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.

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

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

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

123456789

223456789

333456789

433456789

533456789

633456789

Americans

SSN

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.

The inverse of the social security function

Joe

Samantha

Anna

Ian

Chelsea

George

123456789

223456789

333456789

433456789

533456789

633456789

SSN

Americans

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.

.

Domain of f

Range of f

Theorem

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

y = x

(0, 2)

(2, 0)

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

Step1:

Step2: Interchange x and y

Step 3: Solve for y