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Functions
Imagine functions are like the dye you use
to color eggs. The white egg (x) is put in
the function black dye B(x) and the result
is a black egg (y).
The Inverse Function “undoes” what the function
does.
The Inverse Function of the BLACK dye is
bleach.
The Bleach will “undye” the black 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 = 3y2 + 2
x - 2 = 3y
2
x-2
= y2
3
x-2
=y
3
Composition of Functions
Function
Composition
( f g )( x)
This does not say “FOG”
You read this “f composed with
g of x”
Function Composition
( f g )( x)
Another way to write this
is
f ( g ( x))
OR
f[g(x)]
Function Composition
Notation
( f g )( x)
INSIDE function 1st
OUTSIDE function last
Function Composition
( f g )( x)
OR
EX 1: f(x) = x2
g(x) = x + 1
Start with g(x)
and put that
into f(x)
f ( g ( x))
= (x + 1)2
= x2 + 2x + 1
Function Composition
( f g )( x)
EX 2: f(x) = x + 2 g(x) = 4 – x2
Start with g(x)
and put that
into f(x)
= (4 – x2) + 2
= -x2 + 6
Function Composition
EX 3: f(x) = x2 + 1
Start with f(x)
and put that
into g(x)
g(x) = 2x
= 2(x2 + 1)
= 2x2 + 2
Evaluating with
Function Composition (Numbers)
( f g )(3)
EX 4: f(x) = x2 + 1
g(x) = 2x
Start with g(x) & g(3) = 2(3) = 6
find g(3). Put
2 + 1 = 37
f(6)
=
(6)
that answer into
f(x).
Function Composition
f(x) = x2 - 4
g(x) = 4x - 1
a) f[g(-1)] = 21
a) g[f(2)] = -1
a) f[g(a + 1)] = 16a2+24a+5