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# Inverse Functions - PowerPoint PPT Presentation

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.

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## PowerPoint Slideshow about 'Inverse Functions' - mirabel

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• 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.

Let f and g be two functions such that

f (g(x)) = x for every x in the domain of g

and

g(f (x)) = x for every x in the domain of f.

The function g is the inverse of the functionf, 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.

Definition of the Inverse Function

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

### Example

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.

• 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

• 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.

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

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

Example

The equation for the inverse of a function f can be found as follows:

• Verify the function is one-to-one (HLT).

• Replace f (x) by y in the equation for f (x).

• Interchange x and y.

• 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.

• 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.

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 = y

Divide both sides by 6.

6

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

x - 3

f -1(x) =

Rename the function f -1(x).

6

Example

Find the inverse of f (x) = 6x + 3.

• Page 118 #58, 76

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

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