# Combinations of Functions

```Combinations of Functions
Warm Up – Graph the piecewise
function.
3  x, x  0
h x    2
 x  1, x  0
Operations with Functions:
 f  g x  f x  g x

Sum

Difference

Product
 f  g x  f x g x

Quotient
f
f x 
 x  
, where g x   0
g x 
g
 f  g x  f x  g x
Example: Let f(x) = 5x&sup2; -2x +3 and
g(x) = 4x&sup2; +7x -5

Find f + g

Find f - g
f ( x)  g ( x)
f ( x)  g ( x)
(5 x 2  2 x  3)  (4 x 2  7 x  5)
(5 x 2  2 x  3)  (4 x 2  7 x  5)
5x 2  4x 2  2x  7 x  3  5
5x 2  4x 2  2x  7 x  3  5
9x 2  5x  2
x 2  9x  8
Example:
Example: Let f(x) = 5x&sup2; and
and g(x) = 3x – 1.

Find f &middot; g

f x   g x 
f x 
, where g x   0
g x 
5 x 3 x  1
2
5 x 3 x   5 x 1
2
2
15 x  5 x
3
2
Find f/g
2
5x
1
, where x 
3x  1
3
Example:
Example: f(x)=2x + 3 and g(x) = x -7
f
Find  x 
g
2x  3
, x7
x7
g
Find  x 
f
x7
3
, x
2x  3
2
Let’s take a look graphically.
f ( 2)  g ( 4)
Find:
1
f ( x)
+
4
= 5
g ( x)
f ( 1)  g ( 0)
Find:
0
f ( x)
+
-4
=-4
g ( x)
Find:
f ( 2)  g ( 1)
0
f ( x)
-
4
=-4
g ( x)
f (5)  g (0)
Find:
3
f ( x)
-
(- 4)
=7
g ( x)
Find:
f ( 4)  g ( 1)
5
f ( x)
x
4
= 20
g ( x)
Find:
f ( 4 )  g ( 2 )
-3
f ( x)
x
5
= - 15
g ( x)

Find:
g (5)  f ( 3)
6
f ( x)

3
=2
g ( x)
Composition of Functions

A composite function is a combination of
two functions.

You apply one function to the result of
another.



The composition of the function f with the
function g is written as f(g(x)), which is read
as ‘f of g of x.’
It is also known as  f  g x , which is read as
‘f composed with g of x.”
In other words:
 f  g x  f g x
Ex: f(x)=2x + 5 and g(x) = x - 3

You can work out a single “rule” for the
composite function g ( f ( x)) in terms of x.
Find  f  g x  g  f x   g 2 x  5
 2 x  5  3
 2x  2

Do you think
same result?
 f  g x
 f  g x   f x  3
 2 x  3  5
will give you the
NO!
 2x  6  5
 2x 1
REMEMBER g  f x   2 x  2
You Try….
f(x) = 2x + 2
g(x) = (x + 2)2
g  f x
 f  g x
Find:
g 2 x  2   (( 2 x  2)  2) 2
 2 x  4 
2
 4 x  8 x  8 x  16
2
 4 x 2  16 x  16


f (( x  2) 2  2)  2 x 2  4 x  4  2
 2x 2  8x  8  2
 2 x 2  8 x  10
You may need to evaluate a composite
function for a particular value of x.
f x   5  3x and g x   x 2  4. Find  f  g 3.
Method 1:
Work out the
composite
function.
Then substitute
3 for x.
f

 g  x   5  3 x 2  4
 3 x 2  7
f
 g 3  33  7
 27  7
 34
2

You may need to evaluate a composite
function for a particular value of x.
f x   5  3x and g x   x 2  4. Find  f  g 3.
Method 2:
Substitute 3 into
g(x).
Substitute that
value into f(x).
g 3  3  4  13
2
f 13  5  313  34

Now, let’s take a look at it graphically……
Find:
f
g  4 
f ( x)
f ( g 4)  f (4)  5
g ( x)
g 4  4
Find:
g
f ( x)
f  3  0
f  3
g ( x)
g 0  4
f  g (2) 
Find:
f ( x)
f 3  4
g ( x)
g  2  3
Find:
f  f (2)
f ( x)
f  2  0
f 0  1
g ( x)
Find:

g f  g  0
f ( x)
g 0  4

g ( x)
f  4  3
g  3  1
```