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To find the inverse of a function:

To find the inverse of a function:. Change the f(x) to a y. Switch the x & y values. Solve the new equation for y. Ex: g(x)=2x 3. y=2x 3 x=2y 3. Ex: Find the inverse of y = -3x+6. y = -3x+6 x = -3y+6 x-6 = -3y. Ex: f(x)=2x 2 -4 Find the inverse equation. y = 2x 2 -4

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To find the inverse of a function:

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  1. To find the inverse of a function: • Change the f(x) to a y. • Switch the x & y values. • Solve the new equation for y.

  2. Ex: g(x)=2x3 y=2x3 x=2y3

  3. Ex: Find the inverse of y = -3x+6. y = -3x+6 x = -3y+6 x-6 = -3y

  4. Ex: f(x)=2x2-4 Find the inverse equation. y = 2x2-4 x = 2y2-4 x+4 = 2y2

  5. Your Turn!f(x) = 3x2 - 4 x=3y2 - 4

  6. Inverse Functions • Given 2 functions, f(x) & g(x), if f(g(x))=x AND g(f(x))=x, then f(x) & g(x) are inverses of each other. *f -1(x) means “f inverse of x”

  7. Ex: Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are inverses. • Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses. f(g(x))= -3(-1/3x+2)+6 = x-6+6 = x g(f(x))= -1/3(-3x+6)+2 = x-2+2 = x ** Because f(g(x))=x and g(f(x))=x, they are inverses.

  8. Your Turn! Givenf(x) = 7x 2 and g(x) = (x + 2)/7, determine if f(x) and g(x) are inverses. Solution: ** Because f(g(x))=x and g(f(x))=x, they are inverses.

  9. Functions Review • Relation – a mapping of input values (x-values) onto output values (y-values). • Here are 3 ways to show the same relation. x y -2 4 -1 1 0 0 1 1 y = x2 Equation Table of values Graph

  10. Domain is the set of all first members in a relation. Range is the set of all second members in a relation. Domain & Range {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)} Example 4: Domain- Range- D: {1, 2} R: {1, 2, 3}

  11. Example 5: Find the Domain and Range of the following relation: {(a,1), (b,2), (c,3), (e,2)} Domain: {a, b, c, e} Range: {1, 2, 3} Page 107

  12. In order for a relationship to be a function… EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT ONE INPUT CAN HAVE ONLY ONE OUTPUT INPUT (DOMAIN) FUNCTIONMACHINE (RANGE) OUTPUT

  13. Example Which of the following relations are functions? R= {(9,10, (-5, -2), (2, -1), (3, -9)} S= {(6, a), (8, f), (6, b), (-2, p)} T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)} No two ordered pairs can have the same first coordinate (and different second coordinates).

  14. Is this a function? 1. {(2,5) , (3,8) , (4,6) , (7, 20)} 2. {(1,4) , (1,5) , (2,3) , (9, 28)} 3. {(1,0) , (4,0) , (9,0) , (21, 0)}

  15. The Vertical Line Test If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function. Page 117

  16. Determine whether or not the following graphs are functions.

  17. YES! Function? #1

  18. YES! Function? #2

  19. Function? #3 NO!

  20. YES! Function? #4

  21. Function? #5 NO!

  22. YES! Function? #6

  23. Function? #7 NO!

  24. Function? #8 NO!

  25. x y • -2 • -1 • 0 0 • 1 1 x = y2 • Inverse relation – switch the x & y-values. ** The inverse of an equation can be found by switching the x with the y and solving for y. **The inverse of a graph is the reflection of the original graph across the line y = x. ** The inverse of a table can be found by switching the x with the y.

  26. Horizontal Line Test • Used to determine whether a function’s inverse will be a function. • If the original function passes the horizontal line test, then its inverse is a function. And thefunction issaid to bea one-to-one function. • If the original function does notpass the horizontal line test, then its inverse is not a function.

  27. Ex: Graph the function f(x)=x2 and determine whether it is a one-to-one-function. Graph does not pass the horizontal line test, therefore the inverse is not a function. F(x) is not a one-to-one function.

  28. Ex: Graph the function f(x)=2x3 and determine whether it is a one-to-one-function. G(x) is a one-to-one function! Inverse is a function!

  29. Drawing the inverseReflection about y=x

  30. Restricting a Domain • When the inverse of a function is not a function, the domain of the function can be restricted to allow the inverse to be a function. • In such cases, it is convenient to consider “part” of the function by restricting the domain of f(x). If the domain is restricted, then its inverse is a function.

  31. Restricting the Domain Recall that if a function is not one-to-one, then its inverse will not be a function.

  32. Restricting the Domain If we restrict the domain values of f(x) to those greater than or equal to zero, we see that f(x) is now one-to-one and its inverse is now a function.

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