1. ©2001 by R. Villar All Rights Reserved
6.2 Function Operations
©2001 by R. Villar
All Rights Reserved

2. Function Operations
You can perform operations, such as addition, subtraction, multiplication, and division, with functions…
For example:
If f(x) = 3x and g(x) = x – 5
f(x) + g(x) = 3x + (x – 5)
= 4x – 5
f(x) – g(x) = 3x – (x – 5)
= 2x + 5
f(x) • g(x) = 3x(x – 5)
= 3x2 – 15x
What about f(x) ÷ g(x) ?

3. Be sure to consider the domain (the possible inputs)…
f(x) ÷ g(x) = 3x x – 5
Be sure to consider the domain (the possible inputs)…
The domain is does not include 5 since that would make the denominator 0…
Therefore, the domain is all real numbers except x = 5.
Example: Find the domain of
Think of the values that will make the square root a negative number
The domain is all real numbers greater than 1. The graph will confirm this...

4. Composition of Functions
If there is a 40% off sale at Nordstrom’s and as an employee you receive a 10% discount, how much will you pay on a $299 jacket?
You do not get 50% off…
...this is an example of a composite function.
You will pay 90% of the cost (10% discount) after you pay 60% (40% discount).
The two functions look like this… f(x) = 0.6x g(x) = 0.9x
We can put these together in a composite function that looks like this… f(g(x))
“f of g of x”

5. If f(x) = 0.6x and g(x) = 0.9x
What is f(g(x)) when x = 299?
Work from the inside out (find g of 299 first)...
f(g(299)) = f(0.9 • 299)
= f(269.1)
Now, find f of
= 0.6 • 269.1 =
The jacket will cost $161.46

6. Examples: If f(x) = x2 – 5 and g(x) = 3x2 + 1 find f[g(2)] and g[f(2)]
f[g(2)] = f[3(2)2 + 1]
f[g(2)] = f[13]
f[g(2)] = (13)2 – 5
f[g(2)] = 164
g[f(2)] = g[(2)2 – 5]
g[f(2)] = g[ – 1 ]
g[f(2)] = 3(–1)2 + 1
g[f(2)] = 4

7. Example: If f(x) = x2 + 6 and g(x) = 3x – 4 find f(g(x))
f[g(x)] = f(3x – 4)
= (3x – 4)2 + 6
= 9x2 – 24x
= 9x2 – 24x + 22