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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: 1. Determine the domain and range of a function given a graph. 2. Use the vertical line test to determine if a graph is the graph of a function or not. 3. Determine the intervals on which a function is increasing, decreasing or constant by looking at a graph. 4. Determine if a function is even, odd, or neither by looking at a graph. 5. Determine if a function is even, odd, or neither given an equation. 6. 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.

Domain

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.

Range 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

x-intercept No matter what type of graph that you have, recall that the x-intercept is where the graph crosses the x axis. 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.

y-intercept No matter what type of graph that you have, recall that the y-intercept is where the graph crosses the y axis. 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.

Functional Value

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.

Example 1: 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.

a) Domain 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 x values (horizontal axis). 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

b) Range

.

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 y values (vertical axis). 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

.

c) x-intercept If the x-intercept is where the graph crosses the x-axis, what do you think the x-intercept is for this function? If you said x = 3 you are correct. The ordered pair for this x-intercept would be (3, 0).

d) y-intercept If the y-intercept is where the graph crosses the y-axis, what do you think the y-intercept is for this function? If you said y = 3 you are correct. The ordered pair for this y-intercept would be (0, 3).

e) Functional value indicated If the functional value correlates with the second or y value of an ordered pair what is f(2)? 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).

Example 2: 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.

a) Domain 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 x values (horizontal axis). 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

.

b) Range 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 y values (vertical axis). 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

.

c) x-intercept If the x-intercept is where the graph crosses the x-axis, what do you think the x-intercept is for this function? If you said there is none, you are right. Since the graph never crosses the x-axis, then there is no x-intercept.

d) y-intercept If the y-intercept is where the graph crosses the y-axis, what do you think the y-intercept is for this function? If you said y = 3 you are correct. The ordered pair for this y-intercept would be (0, 3).

e) Functional value indicated If the functional value correlates with the second or y value of an ordered pair what is f(-3)? 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).

Example 3: 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.

a) Domain 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 x values (horizontal axis). 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

.

b) Range 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 y values (vertical axis). 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

.

c) x-intercept If the x-intercept is where the graph crosses the x-axis, what do you think the x-intercept is for this function? If you said x = 1 you are correct. The ordered pair for this x-intercept would be (1, 0).

d) y-intercept If the y-intercept is where the graph crosses the y-axis, what do you think the y-intercept is for this function? If you said y = 1 you are correct. The ordered pair for this y-intercept would be (0, 1).

e) Functional value indicated If the functional value correlates with the second or y value of an ordered pair what is f(2)? 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).

Vertical Line Test If no vertical line can be drawn so that it intersects a graph more than once, then it is a graph of a function. 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.

Example 4: Use the vertical line test to identify graphs in which y is a function of x.

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.

Example 5: Use the vertical line test to identify graphs in which y is a function of x.

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.

Increasing

A function is increasing on an interval if for any and in the interval, where , then . In other words, a function is increasing in an interval if it is going up left to right in the entire interval. Below is an example where the function is increasing over the interval . Note how it is going up left to right in the interval .

Decreasing A function is decreasing on an interval if for any and in the interval, where , then . In other words, a function is decreasing in an interval if it is going down left to right in the entire interval. Below is an example where the function is decreasing over the interval

. Note how it is going down left to right in the interval

.

Constant A function is constant on an interval if for any and in the interval, where , then . In other words, a function is constant in an interval if it is horizontal in the entire interval. Below is an example where the function is constant over the interval Note how it is a horizontal line in the interval .

.

Example 6: Use the graph to determine intervals on which the function is a) increasing, if any, b) decreasing, if any, and c) constant, if any.

a) Increasing 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:

b) Decreasing 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:

c) Constant 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:

Example 7: Use the graph to determine intervals on which the function is a) increasing, if any, b) decreasing, if any, and c) constant, if any.

a) Increasing 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:

b) Decreasing 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:

c) Constant 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.

Example 8: Use the graph to determine intervals on which the function is a) increasing, if any, b) decreasing, if any, and c) constant, if any.

a) Increasing 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.

b) Decreasing 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:

c) Constant 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.

Even Function A function is even if for all x in the domain of f . In other words, a function is even if replacing x with -x does NOT change the original function. 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.

Odd Function A function is odd if for all x in the domain of f . In other words, a function is odd if replacing x with -x results in changing every sign of every term of the original function. 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.

Example 9: Determine if the function

is even, odd

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

Even? A function is even if for all x in the domain of f. With that in mind, is this function even? If you said no, you are correct. Note how their second terms have opposite signs, so

.

Odd? A function is odd if that in mind, is this function odd?

for all x in the domain of f. With

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.

Example 10: Determine if the function

is even, odd or

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

Even? A function is even if for all x in the domain of g. With that in mind, is this function even? 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.

Example 11: Determine if the function

is even, odd

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

Even? A function is even if for all x in the domain of f. With that in mind, is this function even? If you said no, you are correct. Note how both of their terms have opposite signs, so

.

Odd? A function is odd if that in mind, is this function odd?

for all x in the domain of f. With

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.

Greatest Integer Function 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.

Example 12: If f(x) = int(x), find the functional value f(7.92). 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

Example 13: If f(x) = int(x), find the functional value f(-3.25). 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.

1a. (answer/discussion to 1a)

1b. (answer/discussion to 1b)

Practice Problems 2a - 2b: Use the vertical line test to identify graphs in which y is a function of x.

2a. (answer/discussion to 2a)

2b. (answer/discussion to 2b)

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.

3a. (answer/discussion to 3a)

3b. (answer/discussion to 3b)

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

4a. (answer/discussion to 4a)

4b. (answer/discussion to 4b)

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

5b

5a. (answer/discussion to 5a)

(answer/discussion to 5b)

5c. (answer/discussion to 5c)

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

f(-9.1)

(answer/discussion to 6a)

Need Extra Help on These Topics?

There are no good web pages to go with these topics. Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

(Back to the College Algebra Homepage) All contents copyright (C) 2002, WTAMU and Kim Peppard. All rights reserved. Last revised on September 25, 2002 by Kim Peppard.