Algebra of Functions
Objectives
To define the sum, difference, product, and
quotient
of functions.
To form and evaluate composite functions.
Basic function operations
Sum 
Difference 
Product 
Quotient 
Function, domain, & range
The domain of a function is the set of all “first
coordinates” of the ordered pairs of a relation.
The range of a function is the set of all “second
coordinates” of the ordered pairs of a relation.
A relation is a function if all values of the domain are
unique (they do not repeat).
A test to see if a relation is a function is the
vertical
line test.
If it is possible to draw a vertical line and cross the graph
of a relation in more than one point, the relation is not
a function.
Example 1
Find each function and state its domain:
Example 2
The efficiency of an engine with a given heat output,
in calories, can be calculated by finding the ratio of
two functions of heat input, D and N, where
| D(i) = i – 5700 and N(i) = i . | ||
 |
Write a function for the efficiency of the engine in terms of heat input (i), in calories. |
 |
 |
Find the efficiency when the heat input is 17,200 calories.
Composition of functions
Composition of functions is the successive
application of the functions in a specific order.
Given two functions f and g, the composite function
is f o g defined by 
and is read
“f of g of x.”
The domain of is f o g the set of elements x in the
domain of g such that g(x) is in the domain of f.
Another way to say that is to say that “the range of function
g must be in the domain of function f.”
A composite function
Example 3
Evaluate ( f o g ) ( x ) and ( g o f ) ( x ) :
f (x) = x − 3
g (x) = 2x2 −1
You can see that function composition is not
commutative!
Example 4
Find the domain of
and 
(Since a radicand can’t be negative in the set of real
numbers, the radicand must be greater than or equal to
zero. This is what limits the domain.)
Example 5
The number of bicycle helmets produced in a factory
each day is a function of the number of hours (t) the
assembly line is in operation that day and is given by
n = P(t) = 75t – 2t2.
The cost C of producing the helmets is a function of
the number of helmets produced and is given by
C(n) = 7n +1000.
Determine a function that gives the cost of producing the
helmets in terms of the number of hours the assembly line is
functioning on a given day.
Find the cost of the bicycle helmets produced on a day
when the assembly line was functioning 12 hours.
(solution on next slide)
C(n) = 7n + 1000
n = P (t ) = 75t − 2t2
Solution to Example 5:
Determine a function that gives the cost of producing
the helmets in terms of the number of hours the
assembly line is functioning on a given day.
Find the cost of the bicycle helmets produced on a day
when the assembly line was functioning 12 hours.
Review
If f ( x ) = 2 x + 1 a n d g ( x ) = x2 , find f (g (x)).
Find g (f (x)).
What is the domain of g (f (x ))?
Consider the functions
and 
Why are their domains different?
Answers to review:
Domain of g (f (x)) is {x : x ∈
}
The domains of the two functions are different because
the denominator of b(x) cannot be zero.
Summary…
Function arithmetic – add the functions (subtract, etc)
Addition
Subtraction
Multiplication
Division
Function composition
Perform function in innermost parentheses first
Domain of “main” function must include range of “inner”
function