**College Algebra**

**Tutorial 32: Graphs of Functions, Part II:**

Domain/Range, Vertical Line Test, Increasing/Decreasing/Constant Functions, Even/Odd Functions, and Greatest Integer Function

**Learning Objectives**

After completing this tutorial, you should be able to:

- Determine the domain and range of a function given a graph.
- Use the vertical line test to determine if a graph is the graph of a function or not.
- Determine the intervals on which a function is increasing, decreasing or constant by looking at a graph.
- Determine if a function is even, odd, or neither by looking at a graph.
- Determine if a function is even, odd, or neither given an equation.
- Apply the greatest integer function to any given number.

**Introduction**

In this tutorial we will take a close look at several
different aspects
of graphs of functions. First we will look at finding the domain
and range of a function given a graph. Next I will show you how a
vertical line can help us determine if a graph is a graph of a function
or not. Then we will look at what it means for a function to be
increasing,
decreasing or constant. This will be followed by showing you how
to tell if a function is even, odd, or neither given either a graph of
the function or just its assignment. We will finish the
lesson
by taking a peek at the greatest integer function. If you need a
review on the definition of a function, feel free to go to **Tutorial
30: Introduction to Functions.** Sounds like we have our
work
cut out for us in this lesson. I guess you better get to it.

** Tutorial**

Let's start by reviewing some terms associated with
functions and how
they pertain to graphs of a function.

Recall that the domain is the set of all input values
to which the
rule applies. These are called your independent variables.
These are the values that correspond to the first components of the
ordered
pairs it is associated with. If you need a review on the domain,
feel free to go to **Tutorial
30: Introduction to Functions.**

On a graph, the domain corresponds to the horizontal
axis. Since
that is the case, we need to look to the left and right to see if there
are any end points to help us find our domain. If the graph keeps going
on and on to the right then the domain is infinity on the right side of
the interval. If the graph keeps going on and on to the left then
the domain is negative infinity on the left side of the
interval.
If you need a review on finding the domain given a graph, feel free to
go to **Tutorial 31: Graphs of
Functions,
Part I.**

Recall that the range is the set of all output
values. These
are called your dependent variables. These are the values that
correspond
to the second components of the ordered pairs it is associated
with.
If you need a review on the range, feel free to go to **Tutorial
30: Introduction to Functions.**

On a graph, the range corresponds to the vertical
axis. Since
that is the case, we need to look up and down to see if there are any
end
points to help us find our range. If the graph keeps going up with no
endpoint
then the range is infinity on the right side of the interval. If
the graph keeps going down then the range goes to negative infinity on
the left side of the interval. If you need a review on finding
the
domain given a graph, feel free to go to **Tutorial
31: Graphs of Functions, Part I**

The word 'intercept' looks like the word
'intersect'. Think
of it as **where the graph intersects the x-axis**.

If you need more review on intercepts, feel free to go
to **Tutorial
26: Equations of Lines.**

The word 'intercept' looks like the word
'intersect'. Think
of it as **where the graph intersects the y-axis**.

If you need more review on intercepts, feel free to go
to **Tutorial
26: Equations of Lines.**

Recall that the functional value correlates with the
second or *y* value of an ordered pair.

If you need a review on functional values, feel free to
go to **Tutorial
30: Introduction to Functions**.

We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the

Since that is the case, we need to look to the left and
right and see
if there are any end points. In

this case, note how there is a left endpoint at *x* = -5 and then the graph goes on and on forever to the right of
-5.

**This means that the domain is **.

We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the

Since that is the case, we need to look up and down and
see if there
are any end points. In this case, note how the graph has a low
endpoint
of *y* = 0 and it has an arrow going up
from
that.

**This means that the range is .**

If the

If you said *x* = 3
you are correct.

**The ordered pair for this x-intercept
would be (3, 0).**

If the

If you said *y* = 3 you are
correct.

**The ordered pair for this y-intercept
would be (0, 3).**

If the functional value correlates with the second or

If you said *f*(2) = 3 ,
then give yourself
a pat on the back. The functional value at *x* = 2 is 3.

**The ordered pair for this would be (2, 3).**

We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the

Since that is the case, we need to look to the left and
right and see
if there are any end points. In

this case, note how there are arrows on both ends of the graph and
no end points. This means that the graph goes on and on forever
in
both directions.

**This means that the domain is .**

We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the

Since that is the case, we need to look up and down and
see if there
are any end points. In this case, note how the graph has a low
endpoint
of *y* = 2 and it has arrows going up from
that.

**This means that the range is .**

If the

If you said there is none, you are right.

Since the graph never crosses the *x*-axis,
then **there is no x-intercept.**

If the

If you said *y* = 3 you are
correct.

**The ordered pair for this y-intercept
would be (0, 3).**

If the functional value correlates with the second or

If you said *f*(-3) = 2 ,
then give yourself
a pat on the back. The functional value at *x* = -3 is 2.

**The ordered pair for this would be (-3, 2).**

We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the

Since that is the case, we need to look to the left and
right and see
if there are any end points. In

this case, note how there are arrows on both ends of the graph and
no end points. This means that the graph goes on and on forever
in
both directions.

**This means that the domain is .**

We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the

Since that is the case, we need to look up and down and
see if there
are any end points. In

this case, note how there are arrows on both ends of the graph and
no end points. This means that the graph goes on and on forever
in
both directions.

**This means that the range is .**

If the

If you said *x* = 1
you are correct.

**The ordered pair for this x-intercept
would be (1, 0).**

If the

If you said *y* = 1 you are
correct.

**The ordered pair for this y-intercept
would be (0, 1).**

If the functional value correlates with the second or

If you said *f*(2) = -1 ,
then give yourself
a pat on the back. The functional value at *x* = 2 is -1.

**The ordered pair for this would be (2, -1).**

Think about it, if a vertical line intersects a graph in
more than one
place, then the *x* value (input) would
associate
with more than one *y* value (output), and
you
know what that means. The relation is not a function.

The next two examples illustrate this concept.

This graph would pass the vertical line test, because
there would not
be any place on it that we could draw a vertical line and it would
intersect
it in more than one place.

**Therefore, this is a graph of a function.**

This graph would not pass the vertical line test
because there is at
least one place on it that we could draw a vertical line and intersect
it in more than one place. In fact, there are a lot of vertical
lines
that we can draw that would intersect it in more than one place, but we
only need to show one to say it is not a function.

The graph below shows one vertical line drawn through our graph that intersects it in two places: (4, 2) and (4, 6). This shows that the input value of 4 associates with two output values, which is not acceptable in the function world.

**Therefore, this is not a graph of a function.**

**A function is increasing on an
interval if for any and in the
interval,
where , then .**

Below is an example where the function is increasing over the interval . Note how it is going up left to right in the interval .

**A function is decreasing on an
interval if for any and in the
interval,
where ,
then .**

Below is an example where the function is decreasing over the interval . Note how it is going down left to right in the interval .

**A function is constant on an interval
if for any and in the
interval,
where ,
then .**

Below is an example where the function is constant over the interval . Note how it is a horizontal line in the interval .

A function is increasing in an interval when it is going up left to right in that interval? With that in mind, what interval, if any, is this function increasing?

**If you said ,
you are correct. **

Note how the function is going up left to right,
starting at *x* = 3 and everywhere to the right of that.

**Below shows the part of the graph that is increasing:**

A function is decreasing in an interval when it is going down left to right in that interval? With that in mind, what interval, if any, is this function decreasing?

**If you said (2, 3), you are right on. **

Note how the function is going down left to right from *x* = 2 to *x* = 3.

**Below shows the part of the graph that is decreasing:**

A function is constant in an interval if it is horizontal in the entire interval. With that in mind, what interval, if any, is this function constant?

**If you said (-5, 2), pat yourself on the back. **

Note how the function is horizontal starting at *x* = -5 all the way to *x* = 2.

**Below shows the part of the graph that is constant:**

A function is increasing in an interval when it is going up left to right in that interval? With that in mind, what interval, if any, is this function increasing?

**If you said ,
you are correct. **

Note how the function is going up left to right,
starting at *x* = -3 and everywhere to the right of that.

**Below shows the part of the graph that is increasing:**

A function is decreasing in an interval when it is going down left to right in that interval? With that in mind, what interval, if any, is this function decreasing?

**If you said ,
you are right on. **

Note how the function is going down left to right from
negative infinity
to *x* = -3.

**Below shows the part of the graph that is decreasing:**

A function is constant in an interval if it is horizontal in the entire interval. With that in mind, what interval, if any, is this function constant?

**If you said it is never constant, pat yourself on the
back. **

Note how the function is never a horizontal line.

A function is increasing in an interval when it is going up left to right in that interval? With that in mind, what interval, if any, is this function increasing?

**If you said it never increases, you are correct. **

Note how the function never goes up left to right.

A function is decreasing in an interval when it is going down left to right in that interval? With that in mind, what interval, if any, is this function decreasing?

**If you said ,
you are right on. **

Note how the function is going down left to right from
negative infinity
to infinity.

**Below shows the part of the graph that is decreasing:**

A function is constant in an interval if it is horizontal in the entire interval. With that in mind, what interval, if any, is this function constant?

**If you said it is never constant, pat yourself on the
back. **

Note how the function is never a horizontal line.

**A function is even if for all x in the domain
of f **

In terms of looking at a graph, an even function is
symmetric with respect
to the *y*-axis. In other words, the
graph
creates a mirrored image across the *y*-axis.

The graph below is a **graph of an even function**.
Note how
it is symmetric about the *y*-axis.

**A function is odd if for all x in the domain
of f **

In terms of looking at a graph, an odd function is
symmetric with respect
to the origin. In other words, the graph creates a mirrored image
across the origin.

The graph below is a **graph of an odd function**.
Note how
it is symmetric about the origin.

To determine if this function is even, odd, or
neither, we need
to replace *x* with -*x* and compare *f*(*x*)
with *f*(-*x*):

A function is even if for all

If you said no, you are correct. Note how their second terms have opposite signs, so .

A function is odd if for all

If you said no, you are
right.

Looking at ,
we see that the signs of the first and third terms of *f*(-*x*)
and -*f*(*x*)
don’t
match, so .

Since we said no for both even and odd, that leaves us
with our answer
to be neither.

**Final answer: The function is neither even nor odd.**

To determine if this function is even, odd, or
neither, we need
to replace *x* with -*x* and compare *g*(*x*)
with g(-*x*):

A function is even if for all

If you said yes, you are
correct. Note how
all of the terms of *g*(*x*)
and *g*(-*x*)
match
up, so .

**Final answer: The function is
even.**

To determine if this function is even, odd, or
neither, we need
to replace *x* with -*x* and compare *f*(*x*)
with *f*(-*x*):

A function is even if for all

If you said no, you are correct. Note how both of their terms have opposite signs, so .

A function is odd if for all

If you said yes, you are
right.

Looking at ,
note how all of the terms of *f*(-*x*)
and -*f*(*x*)
match up, so .

**Final answer: The function is
odd.**

**int( x) **

**Greatest integer that is less than or
equal to x.**

For example, int(5) = 5, int(5.3) = 5, int(5.9) =
5, because
5 is the greatest integer that is less than or equal to 5, 5.3, and
5.9.

**The basic graph of the function f(x)
= int(x) is:**

Note how it looks like steps.

We need to ask ourselves, what is the greatest integer
that is less
than or equal to 7.92?

If you said 7, you are correct.

**Final answer: 7**

We need to ask ourselves, what is the greatest integer
that is less
than or equal to -3.25?

If you said -4, you are correct.

Be careful on this one. We are working with a negative number. -3 is not a correct answer because -3 is not less than or equal to -3.25, it is greater than -3.25.

**Final answer: -4**

** Practice Problems**

These are practice problems to help bring you to the next level.
It will allow you to check and see if you have an understanding of these
types of problems. **Math works just like anything
else, if you want to get good at it, then you need to practice it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.**
In fact there is no such thing as too much practice.

To get the most out of these, **you should work the problem out on
your own and then check your answer by clicking on the link for the answer/discussion
for that problem**. At the link you will find the answer
as well as any steps that went into finding that answer.

Practice Problems 1a - 1b: Use the graph to determine a) the domain, b) the range, c) the x-intercepts, if any d) the y-intercept, if any, and e) the functional value indicated.

Practice Problems 2a - 2b:Use the vertical line test to identify graphs in whichyis a function ofx.

Practice Problems 3a - 3b:Use the graph to determine intervals on which the function is a) increasing, if any, b) decreasing, if any, and c) constant, if any.

Practice Problems 4a - 4b:Use the graph to determine if the function is even, odd, or neither.

Practice Problems 5a - 5c:Determine if the given function is even, odd or neither.

Practice Problem 6a:Iff(x) = int(x), find the given functional value.

** Need Extra Help on these Topics?**

**There are no appropriate webpages
that can assist
you in the topics that were covered on this page.**

**Go to Get Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for some
more suggestions.**

Last revised on June 18, 2010 by Kim Seward.

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