1. Combinations of Functions
Section 1.5
Combinations of
Functions

2. Combinations of Functions
Arithmetic Combinations of Functions
The domain of an arithmetic combination of functions f and g consists
of all real numbers that are common to the domains of f and g. In the
Case of f(x)/g(x), there is the further restriction that g(x)  0.
Sum, Difference, Product and Quotient Functions
1. Sum
2. Difference
3. Product
4. Quotient

3. Combinations of Functions
Try these:
Given
Find:
1. (f + g)(x)
3. (f g)(x)
2. (f – g)(x)
4. (f / g)(x)

4. Combinations of Funcitons
Try these:
Given:
Find:
1. (f + g)(3)
= 8
2. (f – g)(-1)
= - 8
3. (f g)(-2)
= - 35
4. (g / f)(0)
= -1/5

5. Combinations of Functions
Definition of Composition of Two Functions
The composition of the function f with the function g is
The domain of f ◦ g is the set of all x in the domain of g such
that g(x) is in the domain of f.

6. Combinations of Functions
How to from the composite of two functions:
Given:
Step 1: Think of f(x) as being the primary function and g(x) being the
secondary function. To form the composite substitute the
secondary function into the primary function.
Definition of composite
Substitution of 4 – x2 for x in x + 2
Step 2: Simplify the result.

7. Combinations of Functions
How to from the composite of two functions (cont.):
Given:
Step 1: Think of g(x) as being the primary function and f(x) being the
secondary function. To form the composite substitute the
secondary function into the primary function.
Definition of composite
Substitution of x + 2 for x in 4 – x2
Step 2: Simplify the result.

8. Combinations of Functions
How to find the numeric value for the combination of two functions
Given:
Step 1: Find g(2).
Step 2: Take the value for g(2) and
substitute it into f(x).
Step 3: The answer found in step 2 is the
numeric value for

9. Combinations of Functions
How to find numeric values of composite graphically.
Given:
Step 1: Graph both functions on individual graphs.
Step 2: Find g(0)
g(0) = 1
Step 3: Take the value
from Step 2 and
substitute It in for
x in f(x). Find its
value on the graph.
f(1) = -2
f(x) = x2 - 3
g(x) = 2x + 1

10. Combinations of Functions
Try these

11. Combinations of Functions
What you should have learned:
1. To add, subtract, multiply, and divide functions
To find the composition of one function with
another function.