Inverse Functions

Objectives Students will be able to find inverse

functions and verify that two functions are inverse functions of each other.

Students will be able to use the graph of a functions to determine whether functions have inverse functions.

Students will be able to use the horizontal line test to determine if functions are one-to-one.

Find inverse functions algebraically.

Definition of the Inverse Function

Let f and g be two functions such thatf (g(x)) = x for every x in the domain of gand g(f (x)) = x for every x in the domain of f.

The function g is the inverse of the function f, and denoted by f -1 (read “f-inverse”). Thus,

f ( f -1(x)) = x and f -1( f (x)) = x. The domain of f is equal to the range of f -1, and vice versa.

ExampleShow that each function is the inverse of the other:

f (x) = 3x and g(x) = x/3.Solution To show that f and g are inverses of each other, we must show that f (g(x)) = x and g( f (x)) = x. We begin with f (g(x)).

f (x) = 3x

f (g(x)) = 3g(x) = 3(x/3) = x.

Next, we find g(f (x)).

g(x) = x/3

g(f (x)) = f (x)/3 = 3x/3 = x.

Notice how f -1 undoes the change produced by f.

Graphing Inverses

• Inverses are symmetric about the line y=x

• To graph reverse the x and y coordinates.

• Use the symmetry around y = x

• Example; page 2117 - 118 #38, 16, 18

The Horizontal Line Test For Inverse Functions

A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point.

Example

• Does f(x) = x2+3x-1 have an inverse function?

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ExampleSolution:

This graph does not pass the horizontal line test, so f(x) = x2+3x-1 does not have an inverse function.

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Finding the Inverse of a FunctionThe equation for the inverse of a function f can be found as

follows:1. Verify the function is one-to-one (HLT).2. Replace f (x) by y in the equation for f (x).3. Interchange x and y.4. Solve for y. If this equation does not define y as a function of

x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.

5. If f has an inverse function, replace y in step 3 with f -1(x). We can verify our result by showing that f ( f -1(x)) = x and f -1( f (x)) = x.

Find the inverse of f (x) = 6x + 3.Solution Step 1 Replace f (x) by y.

y = 6x + 3

Step 2 Interchange x and y.

x = 6y + 3 This is the inverse function.

Step 3 Solve for y.x - 3 = 6y Subtract 3 from both sides.

x - 3 = y6

Divide both sides by 6.

Step 4 Replace y by f -1(x).

x - 36

f -1(x) = Rename the function f -1(x).

Example

ExamplesExamples

Page 118 #58, 76Page 118 #58, 76

Groups: Pg 117: # 12, 28, 30, Groups: Pg 117: # 12, 28, 30, 39, 41, 6239, 41, 62

Homework: Pg 117: # 1 – 41 Homework: Pg 117: # 1 – 41 odd, 63, 67, 75odd, 63, 67, 75