**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) = 2x^{2} −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 – 2t ^{2}.**

The cost C of producing the helmets is a function of

the number of helmets produced and is given by

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 − 2t ^{2}**

**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 ) = x^{2} , 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