Inverse FunctionsFinding the Inverse

If f(a) = b, then a function g(x) is an inverse of f if g(b) = a.

The inverse of f(x) is typically noted f-1 (x), which is read “f inverse of x” so equivalently, if f(a) = b, then f-1(b) = a

Our inputs and outputs switch places

Important: The raised -1 used in the notation for inverse functions is simply a notation and does not designate an exponent or power of -1.

1.6 Inverse Functions

If f(2) = 4, what do we know about the inverse?

If f(2) = 4, then f-1(4) = 2.

1.6 Inverse Functions

If h-1(6) = 2, what do we know about the original function, h(x)?

1.6 Inverse functions

Using the table below, find and interpret the following:

A. f(60) B. f-1 (60)

1.6 Inverse Functions

t(minutes)

30 50 60 70 90

F(t) (miles)

20 40 50 60 70

1st example, begin with your functionf(x) = 3x – 7 replace f(x) with y

y = 3x - 7Interchange x and y to find the inverse

x = 3y – 7 now solve for y

x + 7 = 3y= y

f-1(x) = replace y with f-1(x)

Finding the inverse algebraically

3

7x

3

7x

2nd example

g(x) = 2x3 + 1 replace g(x) with y

y = 2x3 + 1Interchange x and y to find the inverse

x = 2y3 + 1 now solve for y

x - 1 = 2y3

= y3

= y g-1(x) = replace y with g-1(x)

Finding the inverse

2

1x

3

2

1x

3

2

1x

Not all functions will have an inverse function.

A function must be a one to one function to have an inverse.

To verify if two functions are inverses of on another, you can check the composition of functions with the inverse. If both solutions equal x, they are inverses.

1.6 Inverse Functions

Consider f(x) =

What is the domain?x + 4 > 0x > -4 or the interval [-4, ∞)

What is the range?y > 0 or the interval [0, ∞)

Function with a restricted domain

4x

Now find the inverse: f(x) = D: [-4, ∞) R: [0, ∞) y =

Interchange x and y

x = x2 = y + 4

x2 – 4 = yf-1(x) = x2 – 4 D: [0, ∞) R: [-4, ∞)

Function with a restricted domain

4x4x4x

4y

Finally, let us consider the graphs:f(x) =D: [-4, ∞) R: [0, ∞)

blue graph

f-1(x) = x2 – 4D: [0, ∞) R: [-4, ∞)

red graph

Functions with a restricted domain

4x

2nd exampleConsider g(x) = 5 - x2 D: [0, ∞)

What is the range?

Make a very quick sketch of the graph

R: (-∞, 5]

Function with a restricted domain

Now find the inverse: g(x) = 5 - x2 D: [0, ∞) R: (-∞, 5] y = 5 - x2

Interchange x and y

x = 5 - y2

x – 5 = -y2

5 – x = y2

= y

but do we want the + or – square root?

g-1(x) = D: (-∞, 5] R: [0, ∞)

Function with a restricted domain

x5

x 5

And, now the graphs: g(x) = 5 - x2

D: [0, ∞) R: (-∞, 5]blue graph

g-1(x) = D: (-∞, 5] R: [0, ∞)

red graph

Functions with a restricted domain

x5

A function is one-to-one if each x and y-value is unique

Algebraically it means if f(a)=f(b), then a=b.

On a graph it means the graph passes the vertical and the horizontal line tests.

If a function is one-to-one it has an inverse function.

One-to-one