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
Introduction
Tutorial
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.
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.
If you need a review on functional values, feel free to go to Tutorial 30: Introduction to Functions.
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 .
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 you said x = 3 you are correct.
The ordered pair for this x-intercept would be (3, 0).
If you said y = 3 you are correct.
The ordered pair for this y-intercept would be (0, 3).
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).
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 .
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 you said there is none, you are right.
Since the graph never crosses the x-axis, then there is no x-intercept.
If you said y = 3 you are correct.
The ordered pair for this y-intercept would be (0, 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).
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 .
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 you said x = 1 you are correct.
The ordered pair for this x-intercept would be (1, 0).
If you said y = 1 you are correct.
The ordered pair for this y-intercept would be (0, 1).
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.
Therefore, this is a graph of 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 .
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:
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:
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:
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:
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:
If you said it is never constant, pat yourself on the back.
Note how the function is never a horizontal line.
If you said it never increases, you are correct.
Note how the function never goes up left to right.
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:
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.
If you said no, you are correct. Note how their second terms have opposite signs, so .
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 .
Final answer: The function is neither even nor odd.
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.
If you said no, you are correct. Note how both of their terms have opposite signs, so .
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.
The basic graph of the function f(x) = int(x) is:
Note how it looks like steps.
If you said 7, you are correct.
Final answer: 7
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
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 which y is a function of x.
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: If f(x) = int(x), find the given functional value.
Need Extra Help on these Topics?
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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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