inverse functions

FunctionsFunctionsImagine functions are like the dye you use Imagine functions are like the dye you use to color eggs. The white egg (x) is put in to color eggs. The white egg (x) is put in the function blue dye B(x) and the result is the function blue dye B(x) and the result is a blue egg (y).a blue egg (y).

The Inverse Function “undoes” what the function The Inverse Function “undoes” what the function does.does.

The Inverse Function of the BLUE dye is bleach.The Inverse Function of the BLUE dye is bleach.

The Bleach will “undye” the blue egg and make it The Bleach will “undye” the blue egg and make it white.white.

In the same way, the inverse of a given In the same way, the inverse of a given function will “undo” what the original function will “undo” what the original function did. function did.

For example, let’s take a look at the square For example, let’s take a look at the square function: f(x) = xfunction: f(x) = x22

33

xx f(x)f(x)

3333333333 99999999999999

yy ff--1--1(x)(x)

99999999999999 33333333333333

x2 x

555555555555 252525252525252525252525252525252525252555 5555555555555555

In the same way, the inverse of a given In the same way, the inverse of a given function will “undo” what the original function will “undo” what the original function did. function did.

For example, let’s take a look at the square For example, let’s take a look at the square function: f(x) = xfunction: f(x) = x22

xx f(x)f(x) yy ff--1--1(x)(x)

x2 x

111111111111111111111111 121121121121121121121121121121121121121121121121121121121121121121121121121121121121 11111111111111111111111111111111

In the same way, the inverse of a given In the same way, the inverse of a given function will “undo” what the original function will “undo” what the original function did. function did.

For example, let’s take a look at the square For example, let’s take a look at the square function: f(x) = xfunction: f(x) = x22

xx f(x)f(x) yy ff--1--1(x)(x)

x2 x

Graphically, the x and y values of a Graphically, the x and y values of a point are switched.point are switched.

The point (4, 7)The point (4, 7)

has an inverse has an inverse point of (7, 4)point of (7, 4)

ANDAND

The point (-5, 3)The point (-5, 3)

has an inverse has an inverse point of (3, -5)point of (3, -5)

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Graphically, the x and y values of a point are switched.Graphically, the x and y values of a point are switched.

If the function y = g(x) If the function y = g(x) contains the pointscontains the points

then its inverse, y = gthen its inverse, y = g-1-1(x), (x), contains the pointscontains the points

xx 00 11 22 33 44

yy 11 22 44 88 1616

xx 11 22 44 88 1616

yy 00 11 22 33 44

Where is there a Where is there a line of reflection?line of reflection?

The graph of a The graph of a function and function and

its inverse are its inverse are mirror images mirror images about the line about the line

y = xy = xy = f(x)y = f(x)

y = fy = f-1-1(x)(x)

y = xy = x

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

Inverse Functions

• Given 2 functions, f(x) & g(x), if f(g(x))=x AND g(f(x))=x, then f(x) & g(x) are inverses of each other.

Symbols: f -1(x) means “f inverse of x”

Ex: Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are inverses.

• Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses.

f(g(x))= -3(-1/3x+2)+6

= x-6+6

= x

g(f(x))= -1/3(-3x+6)+2

= x-2+2

= x

** Because f(g(x))=x and g(f(x))=x, ** Because f(g(x))=x and g(f(x))=x, they are inversesthey are inverses..

To find the inverse of a function:To find the inverse of a function:

1. Change the f(x) to a y.

2. Switch the x & y values.

3. Solve the new equation for y.

** Remember functions have to pass the vertical line test!

Ex: (a)Find the inverse of f(x)=x5.

1. y = x5

2. x = y5

3. 5 55 yx

yx 5

5 xy

(b) Is f -1(x) a function?

(hint: look at the graph!

Does it pass the vertical line test?)

Yes , f -1(x) is a 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.

Ex: Graph the function f(x)=x2 and determine whether its inverse is a

function.

Graph does not pass the horizontal line test, therefore the inverse is not a function.

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

yx

3

2

3

2

xy

OR, if you fix the tent in the basement…

2

43 xy

Assignment