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. y = 2x and y = ½x are inverses of each other
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 x 1 2 3 4 y 8 16 x 1 2 4 8 16 y 3

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: Step 1: Switch x and y:
x = 3y2 + 2
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 x 1 2 3 4 y 8 16
then its inverse contains the points x 1 2 4 8 16 y 3
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. Does the graph represent a one-to-one function?x y
Yes No x y

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