1. 6.7 Notes – Inverse Functions

10. Notice how the x-y values are reversed for the original function and the reflected functions.

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

13. Inverse relation – just think: switch the x & y-values. x = y 2 x y 4 -2 1 -1 0 0 1 1 ** the inverse of an equation: switch the x & y and solve for y. ** the inverse of a table: switch the x & y. ** the inverse of a graph: the reflection of the original graph in the line y = x.

14. Ex: Find an inverse of y = -3x+6. Steps: -switch x & y -solve for y y = -3x+6 x = -3y+6 x-6 = -3y

15. 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”

16. Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses. ** Because f(g(x))=x and g(f(x))=x, they are inverses.

17. To find the inverse of a function:

18. Ex: (a)Find the inverse of f(x)=x 5. 1.y = x 5 2.x = y 5 3. (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.

19. Horizontal Line Test Used to determine whether a function’s inverse will be a function by seeing if the original function passes the h hh horizontal line test. If the original function p pp passes the horizontal line test, then its i ii inverse is a function. If the original function d dd does not pass the horizontal line test, then its i ii inverse is not a function.

20. 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.

21. 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…

22. Ex: g(x)=2x 3 Inverse is a function! y=2x 3 x=2y 3 OR, if you fix the tent in the basement…