This document discusses various function operations including addition, subtraction, multiplication, division, and composition of functions. It provides examples of applying each operation to sample functions, noting that the domain of the results depends on the individual functions and operations to avoid undefined values. Composition of functions is defined as applying the outer function to the result of the inner function, and an example demonstrates evaluating a composite function in two steps.

1. 6.3 Function Operations

2. Review: What is a function?
A relationship where every domain
(x value has exactly one unique range
(y value).
 Sometimes we talk about a FUNCTION
MACHINE, where a rule is applied to
each input of x


3. Function Operations
Addition
Multiplica
: f
g ( x)
tion : f g x
Subtractio n : f
Division
f x
:
f
g
g x
x
f x
g x
f x
g x
g x
f x
g x
where g x
0

4. Adding and Subtracting Functions
Let f x
3x
Find f
f
5x
2x
12 .
f
g ( x)
g and f - g
g ( x)
(3 x
8 and g x
8)
4
f x
(2 x
g x
12 )
(3 x
x
8)
f x
(2 x
20
When we look at functions we also want to look at
their domains (valid x values). In this case, the
domain is all real numbers.
g x
12 )

5. Multiplying Functions
2
Let f x
x - 1 and g x
x
1.
Find f g
f x
x
3
g (x)
x
2
(x
x
2
1
1)( x
1)
In this case, the domain is
all real numbers because
there are no values that will
make the function invalid.

6. Dividing Functions
2
Let f x
x - 1 and g x
x
1.
f
Find
g
f x
x
g x
(x
2
x
1)( x
(x
1)
1
1
1)
x
1
In this case, the domain is
all real numbers EXCEPT
-1, because x=-1 would give
a zero in the denominator.

7. Let’s Try Some
Let f x
Find f x
g ( x)
2
5 x - 1 and g x
What is the domain?
5x
1.
Find f x
g ( x)

8. Composite Function – When you
combine two or more functions

The composition of
function g with
function is written as
g f
x
g f x
1
1. Evaluate the inner function f(x) first.
2. Then use your answer as the input of
the outer function g(x).
2

9. Example – Composition of
Functions
Let f x
x
2 and g x
Method 1:
g f
g x
g f
x . Find
5
Method 2:
x
g f
g f x
g(x
g f
2
2)
5
( 7)
(x
5
2
49
2)
2
2
2
g f
x
g f x
5
g( 5
g ( 7)
( 7)
2
49
2)