1. 4-2 OPERATIONS ON
FUNCTIONS
Objectives:
1. Add, subtract multiply and divide functions.
2. Compose functions
3. Determine the domains of resulting functions

2. +, -, ×, ÷ FUNCTIONS
Sum: (f + g)(x) = f(x) + g(x)
Difference: (f – g)(x) = f(x) – g(x)
Product: (f ∙ g)(x) = f(x) ∙ g(x)
Quotient:

3. EXAMPLE 1
Let f(x) = x + 1 and g(x) = x2 – 1, find:
(f + g)(x)


4. YOU TRY!
Let f(x) = x2 + x and g(x) = x + 1, find:
(f – g)(x)
(f ∙ g)(x)

5. ADDING/SUBTRACTING GRAPHS
To find (f + g)(x) or (f – g)(x) from
their graphs, add or subtract the
corresponding y-values.
Example:
f(-2) – g(-2)
f(-4) + g(-4)
f(6) – g(6)
f(x)
g(x)

6. YOU TRY!
Find:
(f + g)(-6)
(f – g)(-2)
(g – f)(1)
(g + f)(5)
f(x)
g(x)

7. FUNCTION COMPOSITION
One function becomes the input of
another function
(f ◦ g)(x) = f(g(x))
 g(x) becomes the input for f
(g ◦ f)(x) = g(f(x))
 f(x) becomes the input for g

8. COMPOSITE FUNCTION DOMAINS
The domain of f ◦ g includes x values
such that:
1. x is in the domain of g
2. g(x) is in the domain of f
So:
 Find domain of input function
 Modify (if needed) so it also works
for the new composite function

9. EXAMPLE 2
Let and
Find a rule for (f ◦ g)(x) and give the domain.

10. EXAMPLE 3
Let and
Find a rule for (f ◦ g)(x) and (g ◦ f)(x)
and give the domain of each.

11. YOU TRY!
Let and
Find (f ◦ g)(x) and its domain.

12. EXTRA EXAMPLE
Let , and
Find h(g(f(4))
Find h(g(f(x))

13. YOU TRY!
Let , and
Find g(h(f(9))
Find g(h(f(x))