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# Inverse functions

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### Inverse functions

1. 1. Inverse Functions Inverse Functions
2. 2. Functions Imagine functions are like the dye you use to color eggs. The white egg (x) is put in the function blue dye B(x) and the result is a blue egg (y).
3. 3. The Inverse Function “undoes” what the function does. The Inverse Function of the BLUE dye is bleach. The Bleach will “undye” the blue egg and make it white.
4. 4. In the same way, the inverse of a given function will “undo” what the original function did. For example, let’s take a look at the square function: f(x) = x 2 3 x f(x) 3 3 3 3 3 9 9 9 9 9 9 9 y f --1 (x) 9 9 9 9 9 9 9 3 3 3 3 3 3 3 x 2
5. 5. 5 5 5 5 5 5 25 25 25 25 25 25 25 25 25 25 5 5 5 5 5 5 5 5 5 In the same way, the inverse of a given function will “undo” what the original function did. For example, let’s take a look at the square function: f(x) = x 2 x f(x) y f --1 (x) x 2
6. 6. 11 11 11 11 11 11 121 121 121 121 121 121 121 121 121 121 121 121 121 121 11 11 11 11 11 11 11 11 In the same way, the inverse of a given function will “undo” what the original function did. For example, let’s take a look at the square function: f(x) = x 2 x f(x) y f --1 (x) x 2
7. 7. Graphically, the x and y values of a point are switched. The point (4, 7) has an inverse point of (7, 4) AND The point (-5, 3) has an inverse point of (3, -5)
8. 8. Graphically, the x and y values of a point are switched. If the function y = g(x) contains the points then its inverse, y = g -1 (x), contains the points Where is there a line of reflection? 16 8 4 2 1 y 4 3 2 1 0 x 4 3 2 1 0 y 16 8 4 2 1 x
9. 9. The graph of a function and its inverse are mirror images about the line y = x y = f(x) y = f -1 (x) y = x
10. 10. Find the inverse of a function : Example 1: y = 6x - 12 Step 1: Switch x and y: x = 6y - 12 Step 2: Solve for y:
11. 11. Example 2: Given the function : y = 3x 2 + 2 find the inverse: Step 1: Switch x and y: x = 3y 2 + 2 Step 2: Solve for y:
12. 12. Ex: Find an inverse of y = -3x+6. <ul><li>Steps: -switch x & y </li></ul><ul><li> -solve for y </li></ul><ul><li>y = -3x+6 </li></ul><ul><li>x = -3y+6 </li></ul><ul><li> x-6 = -3y </li></ul>
13. 13. Inverse Functions <ul><li>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. </li></ul>Symbols: f -1 (x) means “f inverse of x”
14. 14. Ex: Verify that f(x)=-3x+6 and g(x)= -1 / 3 x+2 are inverses. <ul><li>Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses. </li></ul>f(g(x))= -3(- 1 / 3 x+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 .
15. 15. To find the inverse of a function: <ul><li>Change the f(x) to a y. </li></ul><ul><li>Switch the x & y values. </li></ul><ul><li>Solve the new equation for y. </li></ul><ul><li>** Remember functions have to pass the vertical line test! </li></ul>
16. 16. Ex: (a)Find the inverse of f(x)=x 5 . <ul><li>y = x 5 </li></ul><ul><li>x = y 5 </li></ul>(b) Is f -1 (x) a function? (hint: look at the graph! Does it pass the vertical line test?) Yes , f -1 (x) is a function.
17. 17. Horizontal Line Test <ul><li>Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test . </li></ul><ul><li>If the original function passes the horizontal line test, then its inverse is a function . </li></ul><ul><li>If the original function does not pass the horizontal line test, then its inverse is not a function . </li></ul>
18. 18. Ex: Graph the function f(x)=x 2 and determine whether its inverse is a function. Graph does not pass the horizontal line test, therefore the inverse is not a function.
19. 19. Ex: f(x)=2x 2 -4 Determine whether f -1 (x) is a function, then find the inverse equation. f -1 (x) is not a function. y = 2x 2 -4 x = 2y 2 -4 x+4 = 2y 2 OR, if you fix the tent in the basement…
20. 20. Ex: g(x)=2x 3 Inverse is a function! y=2x 3 x=2y 3 OR, if you fix the tent in the basement…
21. 21. Assignment