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SECTION 4.2

SECTION 4.2. ONE-TO-ONE FUNCTIONS INVERSE FUNCTIONS. INVERSE FUNCTIONS. There are some functions which we almost intuitively know as inverses of each other: Cubing a number, taking the cube root of a number. Adding a value to a number, subtracting that value from the number.

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SECTION 4.2

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  1. SECTION 4.2 • ONE-TO-ONE FUNCTIONS • INVERSE FUNCTIONS

  2. INVERSE FUNCTIONS • There are some functions which we almost intuitively know as inverses of each other: • Cubing a number, taking the cube root of a number. • Adding a value to a number, subtracting that value from the number.

  3. INVERSE FUNCTIONS These are inverses of each other because one undoes the other. Can a more complicated function have an inverse? Adds two and divides by 6.

  4. What must the inverse of f(x) do to its variable, x? Multiply by 6 and subtract two.

  5. x f(x) x g(x) Symbolically? 0 1 4 - 2 - 8 1/3 1/2 1 0 - 1 0 1 - 1 1/3 1/2 - 2 4 - 8 0 1 Numerically? Graphically?

  6. f(x) g(x) ANOTHER IMPORTANT OBSERVATION ¯4 ¯ 1 ¯ 1 ¯ 4 g(f(4)) = 4 In fact, the same thing happens for any x-value. g(f(x)) = x

  7. EXAMPLE: • Find the inverse of f(x) which we refer to as f -1(x). • Then, check algebraically to ensure that f(f -1(x)) = x. f -1(x) = (x - 6) 3

  8. Check that f (f -1(x)) = x = x - 6 + 6 = x

  9. RECALL: DEFINITION OF FUNCTION A set of ordered pairs in which no two ordered pairs have the same first coordinate. In other words: FOR EVERY X, THERE IS ONLY ONE Y.

  10. Consider the function f(x) = x 2 x f(x) x y If this function had an inverse, the ordered pairs would have to be reversed. 0 0 0 0 1 1 1 1 2 4 4 2 -1 1 1 -1 4 -2 -2 4

  11. A set of ordered pairs in which no two ordered pairs have the same first coordinate and no two ordered pairs have the same second coordinate. DEFINITION OF ONE-TO-ONE FUNCTION In other words: FOR EVERY X, THERE IS ONLY ONE Y. FOR EVERY Y, THERE IS ONLY ONE X.

  12. f(x) = x 2 is not a one-to-one function. Thus, it has no inverse.

  13. Recall a graphical test which enables us to determine whether a relation is a function. “VERTICAL LINE TEST”

  14. What kind of graphical test would help us to determine whether a function was one-to-one? “HORIZONTAL LINE TEST”

  15. FINDING A FORMULA FOR f -1(x) Example: First of all, check to see if it is one-to-one. Graph it!

  16. Now, find the formula: x y = 2x + 5 x y - 2x = 5

  17. EXAMPLE: Find inverses for the two functions below and graph them to see symmetry. f(x) = 3x - 4 g -1(x) does not exist.

  18. RESTRICTING DOMAINS Example: f(x) = x 2 - 4x f(x) = x 2 - 4x + 4 - 4 f(x) = (x - 2) 2 - 4 Vertex: (2, - 4) Domain: (x ³ 2)

  19. CONCLUSION OF SECTION 4.2

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