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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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Inverse Functions

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Inverse functions l.jpg

Inverse Functions


Objectives l.jpg

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 l.jpg

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


Example l.jpg

Show that each function is the inverse of the other:

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.


Graphing inverses l.jpg

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


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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.


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Example

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


Example8 l.jpg

Solution:

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

Example


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Finding the Inverse of a Function

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.


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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 = 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.


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Examples

  • Page 118 #58, 76

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

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


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