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Inverses of Functions

Inverses of Functions. Section 1.6. Objective. To be able to find inverses of functions. Relevance. To be able to model a set of raw data after a function to best represent that data. Go Over Quiz. Warm Up: Graph & give the domain & range. Answer on Next Slide.

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Inverses of Functions

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  1. Inverses of Functions Section 1.6

  2. Objective • To be able to find inverses of functions.

  3. Relevance • To be able to model a set of raw data after a function to best represent that data.

  4. Go Over Quiz

  5. Warm Up: Graph & give the domain & range. Answer on Next Slide

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

  7. Intro to Inverses • Sneetches Video or Read Story Quickly • A function describes the relationship between 2 variables, applying a rule to an input that generates exactly one output. • For such relationships, we are often compelled to “reverse” or “undo” the rule.

  8. The Sneetches Story • Some of the Sneetches have stars on their yellow bellies; some do not. • In the story, Sylvester McMonkey McBean builds a machine that can apply these stars. • Sneetches without stars pay McBean to make them look like their start-bellied friends. • When all Sneetches appear the same, another problem emerges. So………………

  9. The Sneetches Story • McBean builds a second machine to “reverse” the action of the first one – one that removes the stars. • Can you identify the math connections to this story?

  10. Math Connections functions • McBean’s machine represents____________. • The second machine “reverses” the action of the first, thereby acting as an ____________. inverse

  11. Progress of Inverses Throughout Math • Learned Addition and then its inverse operation Subtraction. • Learned Multiplication and then its inverse operation Division. • Learning Perfect Squares connects with extracting Square Roots • Basically – Inverses are a second operation that reverses the first one!

  12. Calculator Inverses • Take a look at your GDC’s and observe the keys. • Do you notice that inverse operations of many calculator commands are “second” functions? • A calculator key pairs an operation or function with its inverse.

  13. Inverse Function Partner Share. List the sequence of steps needed to evaluate this function: 1. Square the Input 2. Multiply by 9 3. Add 4

  14. Inverse Functions Partner Share • Divide into groups of 3 or 4. • Each group will be given two functions. • For each function: 1) List the steps of each function 2) Provide 3 ordered pairs that will satisfy the function. • Seek out other teams who had the other half of the function share information. For example, a team that had 2A will seek out a team that had 2B. • Class Discussion – What do you notice about the “undoing” of inverses of functions?

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

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

  17. 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!

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

  19. 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

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

  21. Find Inverse:

  22. Find Inverse:

  23. Find Inverse:

  24. Find Inverse:

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

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

  27. How Does an Inverse Exist

  28. More on One-To-One • Recall that a function is a set of ordered pairs where every first coordinate has exactly one second coordinate. • A one-to-one function has the added constraint that each 2nd coordinate has exactly one 1st coordinate.

  29. Example

  30. Example

  31. Example

  32. 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

  33. Reflecting..

  34. 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

  35. Horizontal Line Test • Recall that a function passes the vertical line test. • The graph of a one-to-one function will pass the horizontal line test. (A horizontal line passes through the function in only one place at a time.)

  36. Is it an Inverse? A function can only have an inverse if it is one-to-one. You can use the horizontal line test on graphical representations to see if the function is one-to-one.

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

  38. 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.

  39. Are f & g inverses?

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

  41. Are f & g inverses?

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