Inverse

1. Inverse Inverse

2. Inverse Imagine relations are like the dye you use to color eggs. The white egg is put in the relation blue dye and the result is a blue egg .

3. The Inverse “undoes” what the relation does. The Inverse 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 relation will “undo” what the original relation did. y = 2x and y = ½x are inverses of each other 3 2x 6 3 3 3

5. Domain and Range • This means that • the relation’s domain is the range of the inverse • The relation’s range is the domain of the inverse Relation Inverse

6. Ordered Pairs • Relation • {(0,1),(1,2),(2,4),(3,8),(4,16)} • Inverse • {(1,0),(2,1),(4,2),(8,3),(16,4)}

7. Find the inverse of an Equation Example 1: y = 6x - 12 Step 1: Switch x and y: x = 6y - 12 Step 2: Solve for y:

8. Example 2: y = 3x2 + 2 Find the inverse: x = 3y2 + 2 Step 1: Switch x and y: Step 2: Solve for y:

9. 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)

10. Graphically, the x and y values of a point are switched. If the relation contains the points then its inverse contains the points Where is there a line of reflection?

11. 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

12. Drawing the graph

13. Function Inverse • Every function has an inverse. • The Inverse has to be evaluated separately as to whether it is a function or not. • If f(x) is a function and the inverse of f is also function then the notation f-1(x) can be used for the inverse.

14. Function Inverse • How can I tell if the inverse is a function? • Ordered pairs: write the inverse and make sure no x’s repeat. • Equation: Find the inverse and make sure that the equation will always only produce one answer. • Graph: Create the graph and do the vertical line test.

15. Function Inverse • If the original relation and the inverse is a function we call the function one-to-one. • This means every x creates a unique output. • The graph of the function will then pass the vertical line test as well as the horizontal line test.

16. Yes No x x y y Does the graph represent a one-to-one function?

17. No No x x y y Does the graph represent a one-to-one function?