Inverse functions
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  • 1. Inverse Functions Inverse Functions
  • 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. 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. 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 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. 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. 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. 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. 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. 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. 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. Ex: Find an inverse of y = -3x+6.
    • Steps: -switch x & y
    • -solve for y
    • y = -3x+6
    • x = -3y+6
    • x-6 = -3y
  • 13. 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.
    Symbols: f -1 (x) means “f inverse of x”
  • 14. Ex: Verify that f(x)=-3x+6 and g(x)= -1 / 3 x+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 / 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. 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.
    • ** Remember functions have to pass the vertical line test!
  • 16. Ex: (a)Find the inverse of f(x)=x 5 .
    • y = x 5
    • x = y 5
    (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. Horizontal Line Test
    • Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test .
    • If the original function passes the horizontal line test, then its inverse is a function .
    • If the original function does not pass the horizontal line test, then its inverse is not a function .
  • 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. 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. 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. Assignment