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Inverses of Functions

Inverses of Functions. Warm Up #1 – Find the following. Answer to a. Answer to b. Answer to c. Answer to d. Answer to e. Answer to f. Answer to g. Warm Up #2 - Determine whether the statement is true or false. Justify your answer. FALSE!.

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Inverses of Functions

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  1. Inverses of Functions

  2. Warm Up #1 – Find the following.

  3. Answer to a

  4. Answer to b

  5. Answer to c

  6. Answer to d

  7. Answer to e

  8. Answer to f

  9. Answer to g

  10. Warm Up #2 - Determine whether the statement is true or false. Justify your answer. FALSE!

  11. Warm Up #3: Graph & give the domain & range. Answer on Next Slide

  12. Warm Up #3: Graph & give the domain & range. x y x -3 -4 -5 2 1 0 1 2 3 4 -3 -2 -1 0

  13. Inverse of a relation • The inverse of the ordered pairs (x, y) is the set of all ordered pairs (y, x). • The Domain of the function is the range of the inverse and the Range of the function is the Domain of the inverse. • Symbol: In other words, switch the x’s and y’s!

  14. Example: {(1,2), (2, 4), (3, 6), (4, 8)} Inverse:

  15. Function notation? What is really happening when you find the inverse? Find the inverse of f(x)=4x-2 *4 -2 x 4x-2 (x+2)/4 /4 +2 x So

  16. To find an inverse… • Switch the x’s and y’s. • Solve for y. • Change to functional notation.

  17. Find Inverse:

  18. Find Inverse:

  19. Find Inverse:

  20. Find Inverse:

  21. Draw the inverse. Compare to the line y = x. What do you notice?

  22. Graph the inverse of the following: The function and its inverse are symmetric with respect to the y-axis. x y 0 -3 1 1 -5 -4 2 4

  23. Things to note.. • The domain of is the range of f(x). • The graph of an inverse function can be found by reflecting a function in the line y=x. Check this by plotting y = 3x + 1 and on your graphic calculator. Take a look

  24. Reflecting..

  25. Find the inverse of the function. Is the inverse also a function? Let’s look at the graphs. NOTE: Inverse is NOT a function! Inverse

  26. Horizontal Line Test • If a horizontal line only passes through one point at a time, then the inverse of the function will also be a function.

  27. Composition and Inverses • If f and g are functions and then f and g are inverses of one another. !!!!!!!!!!!!!!!!!!!!!!

  28. Example: Show that the following are inverses of each other. The composition of each both produce a value of x; Therefore, they are inverses of each other.

  29. Are f & g inverses?

  30. You Try…. • Show that • are inverses of each other.

  31. Are f & g inverses?

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