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Bell Work

Bell Work. Given that f(x) = 4x 2/3 and g(x) = 5x 1/2 , 1. Find f(x) + g(x). 2. State the domain of f(x) + g(x). 3 . Find f(g(x)). 6.4 Inverse Functions. Objective: Find inverse functions. In the same way, the inverse of a given function will “undo” what the original function did.

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Bell Work

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  1. Bell Work Given that f(x) = 4x2/3andg(x) = 5x1/2 , • 1. Find f(x) + g(x). • 2. State the domain of f(x) + g(x). • 3. Find f(g(x)).

  2. 6.4 Inverse Functions Objective: Find inverse functions

  3. 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) = x2 f--1(x) y x f(x) 3 9 3 3 9 9 3 3 9 9 3 3 9 9 9 x2 3 3 9 9 3 3 9 9 3 3 9 9

  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) = x2 f--1(x) y x f(x) 5 25 5 5 5 25 25 5 5 5 25 25 5 x2 5 25 5 25 5 25 25 5 25 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) = x2 f--1(x) y x f(x) 121 11 11 11 121 121 11 11 121 121 11 11 121 121 11 x2 121 11 121 11 121 11 121 121 11 11 121 121 11

  6. Vocabulary* • Inverse: • The opposite! • “Undoes” a function DEFINITION: f((g(x)) = x and g(f(x)) = x Notation: f-1(x) is the inverse function of f(x)

  7. To get the inverse we…. Switch the x and y values Try one: y = 7x – 6 Then x = 7y – 6. Then solve for y to get the inverse equation. x + 6 = 7y (x+6)/7 = y

  8. y = f -1(x) Solve for y Trade x and y places Replace f(x) with y *Steps to Remember: Finding the Inverse If it is a function, with f(x), we follow these steps. If it is an equation with y, ignore the first and last step!

  9. Find an equation for the inverse relation. 2. f(x) = (2x3-6)/9

  10. 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.

  11. Graph the function to determine if it has an inverse. • Vocabulary: Horizontal Line Test • If f(x) passes the horizontal line test, it has a inverse function • Do all functions have an inverse? • Do all functions have an inverse function?

  12. 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?

  13. y = f(x) y = x The graph of a function and its inverse are mirror images about the line y = f-1(x) y = x

  14. Graphically, the x and y values of a point are switched. If the x values and y values are switched, what would f(x) = x + 2 and it’s inverse look like? f(x) = x3 and it’s inverse?

  15. Graphically verify that the functions f and g are inverses of each other. If we graph (x - 2)2 it is a parabola shifted right 2. Does this function pass the horizontal line test? This would not pass the horizontal ilne test but they restricted the domain and are only taking the function where x is greater than or equal to 2 so it will pass.

  16. Homework: 6.4 TYPO: 3-10 (1st and 2nd Columns!) 3-10 (1st two columns), 16-20 evens, 22-25 (1st column), 29-40, 42

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