* Introduce functions and function notation * Develop skills in constructing and interpreting the graphs of functions * Learn to apply this knowledge in a variety of situations

1. MATH 1324 – Business College Algebra
3.1 Functions
Chapter 3 Functions and Graphs

2. Concepts and Objectives
The objectives for this section are to
• Introduce functions and function notation
• Develop skills in constructing and interpreting the graphs
of functions
• Learn to apply this knowledge in a variety of situations

3. Functions
• A function consists of a set of inputs called the domain, a
set of outputs called the range, and a rule by which each
input determines exactly one output.
• For example, suppose a rock is dropped straight down
from a high point. From physics, we know that the
distance traveled by the rock in t seconds is 16t2 feet. The
time, t, is the input, and the distance is the output. The
16t2 is the rule. If you think about it, it should be obvious
that each t has only one output associated with it.

4. Functions (cont.)
• Consider a graph of the NASDAQ Composite Index from
January 2011 through January 2017:

5. Functions (cont.)
• Each month has a corresponding data point. The set of
months from January 2011 to January 2017 is the domain
of the function, and the set of all values of the index is the
range.
• Looking at the values for 2015, we can see that the values
of the index duplicate, but this is okay. Different inputs
may produce the same output.

6. Domain and Range
• Recall that the real numbers include all of the whole
numbers, negative numbers, decimals, fractions, and the
irrational numbers such as  or .
• Unless otherwise stated, assume that the domain of any
function defined by a formula or an equation is the largest
set of real numbers that each produces a real number as
output.
2

7. Domain and Range (cont.)
• Example: Find the domain of the following.
a)
b)
4
y x
=
6
y x
= −

8. Domain and Range (cont.)
• Example: Find the domain of the following.
a)
Any number can be raised to the 4th power, so the
domain is all real numbers, written (–, ).
b)
For y to be a real number, x – 6 cannot be negative.
Therefore x  6, or [6, ). (The square bracket [ means
that the interval includes 6.)
4
y x
=
6
y x
= −

9. Functional Notation
• In actual practice, functions are seldom presented in the
style of y = that we have seen. Functions are usually
denoted by a letter such as f. If x is the input variable,
then f(x) denotes the output that f produces from x.
• For example, consider :
( ) 2
1
f x x
= −

10. Functional Notation (cont.)
• To find f (3), or the output produced by the input 3, simply
replace x with 3 in the formula.
• Notice that if we try replacing x with 0, we get
which is not a real number, so therefore, f (0) is not
defined.
( ) 2
3 3 1
9 1 8
f = −
= − =
( ) 2
0 0 1 1,
f = − = −

11. Functional Notation (cont.)
• Any quantity that is a real number that is in the domain
(and produces a real number) can be used, such as a + b
or c4, assuming a, b, and c are real numbers.
( ) ( )
2
2 2
1
2 1
f a b a b
a ab b
+ = + −
= + + −
( ) ( )
2
4 4
8
1
1
f c c
c
= −
= −

12. Piecewise-Defined Function
• To describe some real-world situations, we sometimes
need a function with a multipart rule. Such a function is
called a piecewise-defined function (more informally just
called a piecewise function).
• Example: If you were a single person in Connecticut in
2017 with a taxable income of x dollars and x  $50,000,
then your state income tax T(x) was determined by the
rule
( )
( )
.03 if 0 10,000
300 .05 10,000 if 10,000 50,000
x x
T x
x x
 


= 
+ −  



13. Piecewise-Defined Function
Find the income tax paid by a single person with the given
taxable income.
a) $9200
To find T(9200), since 9200 is less than 10,000, the first
piece of the function applies:
( )
( )
.03 if 0 10,000
300 .05 10,000 if 10,000 50,000
x x
T x
x x
 


= 
+ −  


( ) ( )
9200 .03 9200 $276
T = =

14. Piecewise-Defined Function
b) $30,000
To find T(30,000), since 30,000 is greater than 10,000,
the second piece of the function applies:
( )
( )
.03 if 0 10,000
300 .05 10,000 if 10,000 50,000
x x
T x
x x
 


= 
+ −  


( ) ( )
( )
30000 300 .05 30000 10000
300 .05 20000
300 1000 $1300
T = + −
= +
= + =

15. By Next Class
• 3.1 Functions in MyMathLab
• Quiz 3.1 in Canvas
• Notes on Section 3.2 – Graphs of Functions
• Remember, the classwork and quiz must be completed by
Sunday at 11:59 pm!