Inverse Functions

Inverses ...

Graphically speaking ...

A function can be inverted by switching all of the x's with the corresponding y's in the functions set of ordered pairs.

If this inversion is still a function, one y for every x, it is called an

inverse function f-1

(x)

This is the same as reflecting each point on the functions curve across the line y = x

If point (a, b) lies on the function f(x) then the point (b, a) will lie on the inverse functionf-1

(x)

The domain of the function of f(x) will then become the range of the functionf-1

(x)

Example

f(x) = 2x - 1

f(2 ) = 3 therefore the inverse f-1

(3) = 2

Find the expression for the inverse functionf-1

(x) and then check to see if the above holds true

The steps

1) Test to see if the function is one to one, if it is not the function will not have an invere

2) Replace f(x) by y

3) Interchange x with y

4) Solve for y

5) replace the y with f-1

(x)

Many-to-one functionMore than one member of the domain is mapped to a single member of the range.

1

4

9

-2

5

http://fooplot.com

-2

mapping

outputinputdomain range

Function where each (one) member of the domain is mapped to a unique (one) member of the range.

One-to-one function

1

4

9

-2

5

7

http://fooplot.com

Used only on the graphs of functions. (i.e the vertical line test has already been satisfied).

Horizontal line test

Sweep a horizontal line across the graph of any function if the line crosses the graph more than once the function is many-to-one if the line crosses the graph everywhere exactly once then the function is one-to-one.

For each function find the it's inverse

a) f(x) = 3x

b) g(x) = x - 1

c) f(x) = 2x - 1

d) h(x) = x2

e) f(x) = 2x3 - 4 http://fooplot.com/

e) f(x) = 2x3 - 4 http://fooplot.com/

Example

f(x) = 2x - 1

f(2 ) = 3 therefore the inverse f-1

(3) = 2

Find the expression for the inverse functionf-1

(x) and then check to see if the above holds true

The steps

1) Test to see if the function is one to one, if it is not the function will not have an invere

2) Replace f(x) by y

3) Interchange x with y

4) Solve for y

5) replace the y with f-1

(x)

Find the inverse

Example

If the function f(x) = 5x + 3

find f-1

(4)

Assignment

Exercise 52

1 - 6