The document discusses functions and their inverse functions. It provides examples of functions and their inverses using the square function f(x) = x^2 and explains that the inverse of a function "undoes" what the original function did. It also shows graphically that the inverse function mirrors the original function across the line y = x, switching the x and y values of points. The document demonstrates finding the inverse of two example functions step-by-step.

2. Functions
Imagine functions are like the dye you use
to color eggs. The white egg (x) is put in
the function blue dye B(x) and the result is
a blue egg (y).

3. The Inverse Function “undoes” what the function
does.
The Inverse Function 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
function will “undo” what the original
function did.
For example, let’s take a look at the square
function: f(x) = x2
x
33
33
33
f(x)
y
x2
9999
99
99
9
99
999
f--1(x)
33
33
x 33
3

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
x
55
55
55
f(x)
y
x2
25
25 25
25
25
25
25
25
25
255
f--1(x)
55
55
x 55
55

6. 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
x
f(x)
y
121
11
121
121
11
121 121
11
121 121
11
121 121
x2
11
121
121
121
11
121
121
f--1(x)
11
1
111
11
11
x 11
11
11
1
111

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
10
8
6
x
0
1
2
3
4
y
1
2
4
8 16
2
-10
-8
-6
-4
-2
2
-2
4
6
8
10
4
then its inverse, y = g-1(x),
contains the points
-4
x
1
2
4
8 16
-6
y
0
1
2
3
4
-8
-10
Where is there a
line of reflection?

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

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

11. Example 2:
Given the function : y = 3x2 + 2 find the inverse:
Step 1: Switch x and y: x = 3y2 + 2
Step 2: Solve for y:
x = 3y 2 + 2
2
x − 2 = 3y
x−2
= y2
3
x−2
=y
3