Warm Ups!Warm Ups!

• Find f(g(x)) and g(f(x)) for each of the following:

1. F(x)= 2x +1, g(x) = (x-1)/2

2. F(x) = ½ x + 3, g(x) = 2x-6

Inverse FunctionsInverse Functions

The secret to solving complicated The secret to solving complicated expressionsexpressions

What Is An Inverse?What Is An Inverse?

A function and its inverse “undo” one another, so that the variable is isolated

Details, DetailsDetails, Details

• Domain:– The domain of the

function is the range of the inverse

• Range:– The range of the

function is the domain of the inverse

““Creating” the InverseCreating” the Inverse

Methods :1. Switch the range and

domain value for every point in the function

2. Switch the x (or independent variable) and y (or dependent variable) and solve for y (or dependent variable)

Use:1. When function is

represented as a set of ordered pairs

2. When function is in equation form

Example of Ordered Pairs:Example of Ordered Pairs:

F(x) = { (3,1), (-3, 5), (9,2)}

What is the inverse?F-1(x) = {(1,3),(5,-3),(2,9)}

Example of EquationsExample of Equations1. F(x) = 3x2 +1 2. What is the inverse?3. Y = 3x2 +1 4. X = 3y2 +1 5. X-1 = 3y2

6. (x-1)/3 = y2

7. = y

8. Is this truly a function?• Must restrict the domain, y >0

• Why?• Original range was only positive numbers

How Can I Tell If I’ve Found the How Can I Tell If I’ve Found the Correct Inverse?Correct Inverse?

Graphically:

A function and its inverse are reflections in the line y = x

Algebraically:

If f(f-1(x)) = x AND f-1(f(x)) = x THEN

F-1(x) and f(x) are inverses

Graphically:Graphically:

• A function and its inverse are reflections in the line y = x

Algebraically:Algebraically:

• If f(f-1(x))) = x AND• f-1(f(x)) = x THEN• F-1(x) and f(x) are

inverses

1. F(x)=3x2 +12. f-1(x) = 3. F(f-1(x)) =

3( )2 +1

3((x-1/3) +1x-1 +1x

Now you check if f-1(f(x)) = x

( ) /x 1 3

( ) /x 1 3

ff-1-1(f(x)) = x(f(x)) = x1. F(x)= 3x2 +12. f-1(x) =3. f-1(3x2 +1) =4. f-1 (3x2 +1) =5. f-1(3x2 +1) = 6. f-1(3x2 +1) = x7. Since f-1(f(x)) = x and f(f-1(x))) = x, f and f-1

are inverses

(( ) ) /3 1 1 32x

( ) /x 1 3

(( ) /3 32x

(( )x2