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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?.

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1.8 Inverse Functions, page 222

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  1. 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.

  2. 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.

  3. 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.

  4. 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.)

  5. 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.

  6. See Example 2, page 226.Check Point 2. • Find the inverse of f(x) = 2x + 7.

  7. See Example 3, page 226.Check Point 3. • Find the inverse of f(x) = 4x3 – 1.

  8. See Example 4, page 227.Check Point 4.

  9. 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)!

  10. Horizontal Line TestSee Example 5, page 228. • Check Point 5 – See page 229.

  11. Graphing the Inverse – A reflection about the line y = x. (Swap each x & y.) • See Example 6, page 229.

  12. 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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