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).
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.
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
99
9
9 99
9
9
99
9
9
99
f--1(x)
3
3
3
3
x 333
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
2525
25 25
25
25
2525
25
255
f--1(x)
x
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) = x2
x
11
11
11
11
11
11
f(x)
y
x2
121
121
121
121
121
121
121
121
121
121
121
121
121
121
f--1(x)
11
11
11
11
x 1111
11
11
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)
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
4
y
1
2
4
8
16
2
-10 -8
-6
-4
-2
2
4
6
8
10
then its inverse, y = g-1(x),
contains the points
-2
-4
x
1
2
4
8
16
-6
y
0
1
2
3
4
-8
-10
Where is there a
line of reflection?
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)
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
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
x  2  3y
2
x2
 y2
3
x2
y
3