2. Review 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

3. 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.

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 99 3 3 3 3 3 3 3 x2x2

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) x2x2

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) x2x2

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 x01234y124816 x124816y01234 Where is there a line of reflection?

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. To find the inverse of a function: 1.Change the f(x) to a y. 2.Switch the x & y values. 3.Solve the new equation for y. ** Remember functions have to pass the vertical line test!

11. Find the inverse of a function : Example 1: y = 6x - 12 Example 1: y = 6x - 12 Step 1: Switch x and y: x = 6y - 12 Step 2: Solve for y:

12. 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:

13. Does This Have An Inverse which is a function? Given the function at the right Can it have an inverse? Why or Why Not? NO … when we reverse the ordered pairs, the result is Not a function We would say the function is not one-to-one A function is one-to-one when different inputs always result in different outputs xY 15 29 46 75

14. One-to-One Functions When different inputs produce the same output Then an inverse of the function does not exist (is not a function) When different inputs produce different outputs Then the function is said to be “one-to-one” Every one-to-one function has an inverse which is a function Contrast

15. One-to-One Functions Examples Horizontal line test?

16. 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.

17. 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.

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

19. 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”

20. 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.

21. practice worksheet