Math 251: Beginning Algebra
Section 3.6 Notes
Introduction to Functions
When the elements in one set are linked to elements in a second set, we call this a relation.
Animal
Dog
Cat
Duck
Lion
Rabbit
Set of Inputs=Domain
Life Expectancy
(years)
11
10
7
Set of Outputs=Range
If x is an element in the domain and y is an element in the range, and if a relation exists between x and
y, then we say that y depends on x, and we write x → y . We can also represent this relation as a set of
ordered pairs ( x, y ) , where x represents the input and y represents the output:
{(Dog,11), (Cat,11), (Duck,10), (Lion,10), (Rabbit,7)}
x
y
Example 1:
Set A
Set B
1
5
8
10
12
3
Domain
4
6
0
Range
Represent this relation as a set of ordered pairs ( x, y ) , where x represents the input and y represents the
output:
Identify the domain and range of this relation:
Any set of ordered pairs is a relation.
Math 251: Beginning Algebra
Section 3.6 Notes
Example 2:
Identify the domain and range of the relation: {(2,4), (2, -3), (1, 5)}
A function is a special relation.
It is a set of ordered pairs in which each input corresponds to exactly one output.
Example 3: Determine whether each relation is a function.
(a)
Animal
Dog
Cat
Duck
Lion
Rabbit
(b)
Life Expectancy
(years)
11
10
7
Domain
Range
1
5
8
10
12
3
(c) {(-2,8), (-1,1), (0,0), (1,1), (2,8)}
(d) {(5,2), (5,1), (3,4)}
4
6
0
Math 251: Beginning Algebra
Section 3.6 Notes
Most useful functions have an infinite number of ordered pairs and are usually defined with equations
that tell how y depends on x.
Everyday Examples ☺
1. The distance d a car moving at 45 mph travels is a function of the time t:
d = 45 t
2. The cost y in dollars charged by an express mail company is a function of the weight x in pounds
determined by the equation:
y = 1.5( x − 1) + 9
One way to determine if a relation is a function is to look at the graph of the equation!
y
y
x
x2 + y2 = 1
y = x2
y
x
y = x+2
x
Math 251: Beginning Algebra
Section 3.6 Notes
Vertical Line Test
Intersects
in one point
Intersects in more
than one point
Passes the test
Function
Fails the test
Not a Function
If a vertical line intersects a graph in more than one point,
then the graph is not the graph of a function.
Example 4: Determine whether each relation is a function.
(a)
(c)
(b)
(b)
(d) x = 5
Math 251: Beginning Algebra
Section 3.6 Notes
Function Notation
x
f
f(x)=y
The letters _____, _____, and _____ are commonly used to name functions.
For example, since the equation y = 4 x − 3 describes a function, we may use function notation:
f ( x) = 4 x − 3
Say: “f of x”
For functions, the notations y and f (x) can be used interchangeably.
For the function defined by f ( x) = 4 x − 3 , if x = 2 , then f (2) = _____________________.
The statement f (2) = 5 says that the value of y is _______ when x is _______.
Say: “f of 2 equals 5”
The statement f (2) = 5 also indicates that the point (
,
) lies on the graph of f.
Math 251: Beginning Algebra
Example 5: For each function f, find f (2) , f (−1) , and f (0) .
(a) f ( x) = x − 1
f ( 2) =
f (−1) =
f ( 0) =
(b) f ( x) = x 2 − x + 1
f ( 2) =
f (−1) =
f ( 0) =
Section 3.6 Notes