
Skills to Master
·
Determine
whether a given equation represents a function
·
Determine
whether a given graph represents a function (vertical line test)
·
Understand and
use function notation
·
Graph functions
by creating a table of values
·
Find the vertex
of a quadratic function
·
Graph a
quadratic function by finding the vertex and creating a table of values
Functions and Function Notation
In the previous chapter
you learned about linear equations. This
chapter develops an idea that is important in all areas of mathematics. To begin, consider the following sets of
numbers:
{1, 2, 3, 4} and {2, 4,
6, 8}
There are numerous ways
to take one number from the first set and one from the second set such that
each pair is related in some way. For
example, (1, 2), (2, 4), (3, 6), (4, 8).
In this case, the relation is that the second number is twice the first. There can be many other ways to pair these
numbers, each of which forms a relation.
Another example might be (1, 8), (2, 6), (3, 4), (4, 2). How might you describe this relation?
A function is a specific kind of relation. Think of a function as a machine that takes
an input from one set and outputs something belonging to another set (or
possibly the same set). You’ve seen this
with linear functions. For example, if
our function is
, and our input (for x) is -1, then the output is
. From this you can write
an ordered pair, (-1, -1), and then
you can plot it on the coordinate plane.
What makes a function
special is that, for every input value, there must be exactly one output
value.
NON-EXAMPLE
Suppose we had the following pairs of numbers
for some relation: (4, 1), (3, 7), (3, 1), (2, 1)
In this case, the input
values are the first elements in the ordered pairs and the output values are
the second elements. Notice that for the
input value 3, there are two possible output values: 7 and 1. Hence, this relation is NOT a function.
A function f is a
correspondence between a set of input values x (called the domain)
and a set of output values y
(called the range), where to
each x-value in the domain there
corresponds exactly one y-value
in the range.
In the first example,
the set {1, 2, 3, 4} is the domain and the set {2, 4, 6, 8} is the range.
To determine whether a
given equation represents a function, ask yourself if, for any input value it
is possible for there to be more than one output value.
EXAMPLE
Which of the following
are functions?
![]()
If we input a 3 for x, for instance, can there be more than
one value for y that makes the
equation true?
![]()
If we input a 1 for x, is can there be more than one value
for y that makes the equation true?
![]()
If we input a 2 for x, can there be more than one value for y that makes the equation true?
FUNCTION NOTATION
In chapter 2 we wrote
linear functions in the form
, which assigns the output value directly to y.
More generally, we can say that
, where
(read “f of x”) is a
special notation. It indicates that for
the function f (whatever it may be),
the input values belong to x. So for a linear function, we would write
. This is exactly
equivalent to the above since
. The x can be replaced by any particular
value or even by another variable or another function.
EXAMPLES
For the function
, find
![]()
![]()
![]()
![]()
PRACTICE
For the function
, find
![]()
![]()
![]()
![]()
DOMAIN AND RANGE
For different functions
it is often necessary to determine what the domain and range values can
be. For our purposes, both the domain
and range will come from the set of real numbers. But we will sometimes find restrictions on
which values can be input and limits on which values will be output.
Determining domain is
not too difficult in most cases, but finding the range can sometimes be
challenging. One method is to solve an
equation for x and determine the
range for y in the same way that we
determine the domain for x. But this method doesn’t work for radical
functions (you’ll see why in the example below). In the case of quadratic functions, since the
vertex of the parabola is the maximum (or minimum) value for the function, then
the domain is less than or equal to (or greater than or equal to) the
y-coordinate of the vertex (see last example below).
EXAMPLES
Find the domain and
range for the given functions:
![]()
domain =
range =
![]()
domain =
range =
![]()
domain =
range =
![]()
domain =
range =
PRACTICE
![]()
domain =
range =
![]()
domain =
range =
![]()
domain =
range =
![]()
domain =
range =
To determine domain:
·
If the equation
is linear or quadratic, then the domain is all real numbers.
·
If the equation
contains x in the denominator,
determine which value(s) will result in zero in the denominator; the domain
will have the form
, where c is some real
number.
·
If the equation
contains x inside a square root sign,
determine what values of x inside the
radical will result in values greater than or equal to zero. The domain will have the form
, where c is some real
number
To determine the range:
·
Solve the
equation for x
·
Use the steps
above, substituting y in place of x
·
For a radical
function, solve for the radical if necessary and set the right side of the
function to greater than or equal to zero and solve for y.
·
For quadratic
functions, find the y-coordinate of the vertex and determine whether the
function opens up or down. If it opens
up, then all y values will be greater than or equal to the y-coordinate
value. If it opens down, all y-values
will be less than or equal to the y-coordinate value
THE VERTICAL LINE TEST
A simple way to
determine whether a relation is a function from its graph is by use of the
vertical line test. To use this test,
draw some vertical lines through the graph.
If there is any place on the graph for which the vertical line touches
two or more points on the graph, then the relation is NOT a function.
Example
Quadratic Functions
A quadratic function is
a second-degree polynomial of the form
, where a, b, and c are real numbers and a
is not zero. The graph of a quadratic
equation is a parabola.
Example: The function is ![]()
|
x |
y |
|
-2 |
4 |
|
-1.5 |
2.25 |
|
-1 |
1 |
|
-0.5 |
0.25 |
|
0 |
0 |
|
0.5 |
0.25 |
|
1.5 |
2.25 |
|
2 |
4 |

The easiest way to
graph a quadratic function is to create a table of values and then plotting the
points. The table above shows the values
for the given function.
Characteristics of a Parabola
Some characteristics of
parabolas can be determined from its equation:
·
Vertical
orientation: if the value of a is
greater than zero (positive), then the parabola opens up; if the value of a
is less than zero (negative), then the parabola opens down.
·
Vertex: the x-coordinate of the vertex (the point at which the graph turns) is determined by the
formula
; the y-coordinate can
be determined by substituting the value obtained for x back into the original equation.
·
If the graph intersects
the x-axis, the point or points of
intersection are called the roots or
zeros of the function. Since the points are located on the x-axis, then the values of the
y-coordinates are zero. That is, the
zeros can be found by setting
and solving. Notice that there are either 2, 1, or zero
solutions.
·
Line of Symmetry: the line of symmetry is at
, where c is the value
of the x-coordinate of the vertex. This
line divides the parabola into two mirror-image halves. Points on a horizontal line through the
parabola are equidistant from the line of symmetry.
·
y-intercept: the value
of c is the y-intercept, the point at
which the graph crosses the y-axis.
Example
Find the vertex, zeros,
and line of symmetry for the given function.
Also determine whether the graph opens up or down: ![]()
Vertex:
Line of symmetry:
Zeros:
y- intercept:
Orientation:
These characteristics
can help us graph a parabola.
In general, it’s probably easiest to first find the vertex of the
parabola, then pick 2 x-values on either side of the line of symmetry and create
a table of values.
Example
Graph the quadratic function ![]()
Find the vertex and plot the point on the graph.

Create a table of values, choosing values for x on either side of
the line of symmetry.
|
x |
y |
|
|
|
|
|
|
|
|
|
|
|
|
Plot these points and draw the graph
Try this one:
Graph the quadratic function ![]()
