1 8 inverse functions page 222
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1.8 Inverse Functions, page 222. Objectives Verify inverse functions Find the inverse of a function. Use the horizontal line test to determine one-to-one. Given a graph, graph the inverse. Find the inverse of a function & graph both functions simultaneously. What is an inverse function?.

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1.8 Inverse Functions, page 222

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1 8 inverse functions page 222

1.8 Inverse Functions, page 222

  • Objectives

    • Verify inverse functions

    • Find the inverse of a function.

    • Use the horizontal line test to determine one-to-one.

    • Given a graph, graph the inverse.

    • Find the inverse of a function & graph both functions simultaneously.


What is an inverse function

What is an inverse function?

  • A function that “undoes” the original function.

  • A function “wraps an x” and the inverse would “unwrap the x” resulting in x when the 2 functions are composed on each other.

  • The domain of f is equal to the range of f -1 and vice versa.


How do their graphs compare

How do their graphs compare?

  • The graph of a function and its inverse always mirror each other through the line

    y = x.

  • Example: y = (1/3)x + 2 and its inverse = 3(x - 2)

  • Every point on the graph (x,y) exists on the inverse as (y,x) (i.e. if (-6,0) is on the graph, (0,-6) is on its inverse.


See example 1 page 225 check point 1

See Example 1, page 225.Check Point 1

  • Show that each function is the inverse of the other.

    (Verify by showing f(f -1(x)) = x and f -1(f(x)) = x.)


How do you find an inverse pg 226

How do you find an inverse? (pg 226)

  • Replace f(x) with y.

  • Interchange (swap) the x and the y in the equation.

  • Solve for y.

  • If f has an inverse, replace y in step 3 with f -1(x).

  • Verify.


See example 2 page 226 check point 2

See Example 2, page 226.Check Point 2.

  • Find the inverse of f(x) = 2x + 7.


See example 3 page 226 check point 3

See Example 3, page 226.Check Point 3.

  • Find the inverse of f(x) = 4x3 – 1.


See example 4 page 227 check point 4

See Example 4, page 227.Check Point 4.


Do all functions have inverses

Do all functions have inverses?

  • Yes, and no. Yes, they all will have inverses, BUT we are only interested in the inverses if they ARE A FUNCTION.

  • DO ALL FUNCTIONS HAVE INVERSES THAT ARE FUNCTIONS? NO.

  • Recall, functions must pass the vertical line test when graphed. If the inverse is to pass the vertical line test, the original function must pass the HORIZONTAL line test (be one-to-one)!


Horizontal line test see example 5 page 228

Horizontal Line TestSee Example 5, page 228.

  • Check Point 5 – See page 229.


Graphing the inverse a reflection about the line y x swap each x y

Graphing the Inverse – A reflection about the line y = x. (Swap each x & y.)

  • See Example 6, page 229.


Check point 6

y

5

x

-5

-5

5

Check Point 6

  • The graph of function f consists of two line segments, one segment from (-2, -2) to (-1, 0) and a second segment from (-1, 0) to (1,3). Graph f and use the graph to draw the graph of its inverse function.


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