INVERSE

FUNCTIONS

Set X Set Y

1

2

3

4

5

2

10

8

6

4

Remember we talked about functions---taking a set X and mapping into a Set Y

An inverse function would reverse that process and map from SetY back into Set X

1

2

3

4

5

2

10

8

6

4

123

4

5

2

864

If we map what we get out of the function back, we won’t always have a function going back!!!

Recall that to determine by the graph if an equation is a function, we have the vertical line test.

If a vertical line intersects the graph of an equation more than one time, the equation graphed is NOT a function.

This is a functionThis is NOT a

function This is a function

If the inverse is a function, each y value could only be paired with one x. Let’s look at a couple of graphs.

Look at a y value (for example y = 3)and see if there is only one x value on the graph for it.

For any y value, a horizontal line will only intersection the graph once so will only have one x value

Horizontal Line Test to see if the inverse is a function.

If a horizontal line intersects the graph of an equation more than one time, the equation graphed is NOT a one-to-one function and will NOT have an inverse function.

This is a one-to-one

function

This is NOT a one-to-one

function

This is NOT a one-to-one function

Horizontal Line TestHorizontal Line Test

• Used to determine whether a function’s inverseinverse will be a function by seeing if the original function passes the horizontal line horizontal line testtest.

• If the original function passespasses the horizontal line test, then its inverse is a functioninverse is a function.

• If the original function does not passdoes not pass the horizontal line test, then its inverse is not a inverse is not a functionfunction.

Steps for Finding the Inverse of a One-to-One Function

Replace f(x) with y

Trade x and y places

Solve for y

y = f -1(x)

Find the inverse of

Replace f(x) with y

Trade x and y places

Solve for y

y = f -1(x)

x

xf

2

4

xy

2

4

yx

2

4

42 yx42 xyx

xxy 24

x

xy

24

x

xxf

241

Let’s check this by doing 1ff

xx

ff42

2

41

xxx 422

4

xYes!

x

xxf

421

or

Ensure f(x) is one to one first. Domain may need to be restricted.

Find the inverse of a function :Find the inverse of a function :

Example 1: Example 1: y = 6x - 12y = 6x - 12

Step 1: Switch x and y:Step 1: Switch x and y: x = 6y - 12x = 6y - 12

Step 2: Solve for y:Step 2: Solve for y: x 6y 12

x 12 6y

x 12

6y

1

6x 2 y

Example 2:Example 2:

Given the function : Given the function : y = 3xy = 3x22 + 2 + 2 find the inverse: find the inverse:

Step 1: Switch x and y:Step 1: Switch x and y: x = 3yx = 3y22 + 2 + 2

Step 2: Solve for y:Step 2: Solve for y: x 3y2 2

x 2 3y2

x 23

y2

x 2

3y

Ex: Find an inverse of y = -3x+6.• Steps: -switch x & y

-solve for y

y = -3x+6

x = -3y+6

x-6 = -3y

yx

3

6

23

1

xy

Finding the Inverse

Try2

2

xy

x

1

Given ( ) 2 7

then 2 7

7solve for x x

27

2

f x x

y x

y

yf y

Review from chapter 2Review from chapter 2

• Relation – a mapping of input values (x-values) onto output values (y-values).

• Here are 3 ways to show the same relation.

y = x2 x y

-2 4

-1 1

0 0

1 1

Equation

Table of values

Graph

• Inverse relation – just think: switch the x & y-values.

x = y2

xy

x y

4 -2

5 -1

0 0

1 1

** the inverse of an

equation: switch the x & y and solve for

y. ** the inverse of a

table: switch the x & y.

** the inverse of a graph: the reflection of

the original graph in the line y = x.

42 xy

Consider the graph of the function 42)( xxf

The inverse function is2

4)(1 x

xf

42 xy

2

4

xy

42 xy

2

4

xy

Consider the graph of the function 42)( xxf

The inverse function is2

4)(1 x

xf

An inverse function is just a rearrangement with x and y swapped.

So the graphs just swap x and y!

)4,0(x

)0,4(x

)2,3( x

)3,2( x

42 xy

2

4

xy

)4,0(x

)0,4(

x

)2,3( x

)3,2( x

is a reflection of in the line y = x)(1 xf )(xf

xy What else do you notice about the graphs?

)4,4( x

The function and its inverse must meet on y = x

Graph f(x) and f -1(x) on the same graph.

Graph f(x) and f -1(x) on the same graph.

Let’s consider the function and compute some values and graph them.

3xxf

x f (x)

-2 -8-1 -1 0 0 1 1 2 8

Is this a function? Yes

What will “undo” a cube? A cube root

31 xxf

This means “inverse function”

x f -1(x)

-8 -2-1 -1 0 0 1 1 8 2

Let’s take the values we got out

of the function and put them into the inverse function

and plot them

Notice that the x and y values traded places for the function

and its inverse.

These functions are reflections of each other

about the line y = x

3xxf

31 xxf

(2,8)

(8,2)

(-8,-2)

(-2,-8)

Graph f(x) = 3x − 2 and using the same set of axes.

Then compare the two graphs.

Determine the domain and range of the functionand its inverse.

€

f −1 =x + 2

3

Verify that the functions f and g are inverses of each other.

2;2,2 2 xxgxxxf

If we graph (x - 2)2 it is a parabola shifted right 2.

Is this a one-to-one function?

This would not be one-to-one but they restricted the domain

and are only taking the function where x is greater

than or equal to 2 so we will have a one-to-one function.

e.g. On the same axes, sketch the graph of

and its inverse.

2,)2( 2 xxy

)0,2(

)1,3(

xy

)4,4(x

Solution:

)2,0(

)3,1(

Ex: f(x)=2x2-4 Determine whether f -1(x) is a function, then find the inverse equation.

2

2

4y

x

f -1(x) is not a function.

y = 2x2-4

x = 2y2-4

x+4 = 2y2

2

4x

y

22

1 xyOR, if you fix

the tent in the basement…

Ex: g(x)=2x3

Inverse is a function!

y=2x3

x=2y3

3

2y

x

yx3

2

3

2

xy

OR, if you fix the tent in the

basement…2

43 xy

Exercise

(d) Find and write down its domain and range.

1 (a) Sketch the function where.)(xfy

(e) On the same axes sketch .

)(1 xf

1)( 2 xxf

)(xf

)(1 xf

)(1 xfy

(c) Suggest a suitable domain for so that the inverse function can be found.

(b) Write down the range of . )(xf

12 xy(a)

Solution:

0x( We’ll look at the other

possibilityin a minute. )

Rearrange: 21 xy xy 1

Swap: yx 1

Let 12 xy(d) Inverse:

1)(1 xxf

Domain: 1x Range: 0y

0x(c) Restricted domain:

(b) Range of :)(xf1)( xf

12 xy

Solution:

(a)

Rearrange: 21 xy (d) Let 12 xyAs before

(c)0x

Suppose you chose

for the domain

We now need sincexy 1 0x

(b) Range of :)(xf1)( xf

12 xy

Solution:

(a)

Swap: yx 1

1)(1 xxf

Range:(b) 1y

Domain: 1x Range: 0y

(c)0x

Suppose you chose

for the domain

Rearrange: 21 xy (d) Let 12 xyAs before

We now need sincexy 1 0x

Choosing is easier!

0x