Name ___________________________ 2-5 NotesIB Math SL Date:

Lesson 2-5 Inverse FunctionsLearning objectives:

Find the inverse of a given algebraic function.Find the inverse of a function from the graph

Recall that a function can be represented by a set of ordered pairs. For instance, the function

f ( x )=x+3 from A={1 ,2,3 ,4 } to B={4 ,5 ,6 ,7 } can be written as

f :

By interchanging the x and y coordinates of each ordered pair, you can form the inverse function of f . This function is denoted by f−1. It is a function from B to A, and can be written as

f−1:

1. Show that the functions are inverse functions of each other.

f ( x )=2x3−1 and g ( x )=3√ x+12

Here are some important observations about inverse functions.

If g is the inverse function of f , then f is the inverse function of _______.

The domain of f−1is equal to the _____________ of f , and the range of f−1 is equal to the _____________ of f .

2. If f ( x )= x+42 and g ( x )=2 x−4, evaluate the expressions below.

a. g(1) and f(g(1))

b. f(-3) and g(f(-3))

c. f(g(x))

d. g(f(x))

What can you conclude?

Finding an inverse algebraically.

3. Find the inverse function of

f ( x )=√2 x−3 .

4. Determine the inverse function, f−1 (x ) of f ( x )=2x+3.

Find the inverse function for each of these functions.

5. f (x)=3x−1

6. f ( x )= 3√x−3

7. g ( x )=2 x3+3

8. g ( x )=2x−1

9.

The Existence of an Inverse Function

Not all functions have inverses. A function has an inverse if it is a one-to-one function, i.e. if it passes the horizontal line test. If a horizontal line crosses the graph of a function more than once, then that function does not have an inverse function.

10. Which of these functions have inverse functions?

a.

b.

c.

Sketching the Graph of an Inverse Function

We will be interested in graphing the inverse function, given the graph . The graph of the inverse function is a reflection of the original function in the line y=x . Here are some graphs of functions and their inverse functions:

To reflect a point over the line y=xyou ______________________________________________________.

For example, if the function f contains the points (3, 7), (2, 1), (5, -4), what would be the corresponding points on the graph of f−1 ?

For each of the following, graph the inverse function:

11.

What is the domain and range of f(x)? What is the domain and range of f-1(x)?

12.

13.

Steps:1. Graph the line f(x)=x2. Identify points of the

function f(x).3. Use those points to

identify points of the inverse f−1 (x ) .

14.