Intermediate Algebra
Tutorial 13: Introduction to Functions

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Learning Objectives
After completing this tutorial, you should be able to:
 Know what a relation, function, domain and range is.
 Find the domain and range of a relation.
 Identify if a relation is a function or not.
 Know how to use the vertical line test.
 Evaluate functional values.

Introduction
In this tutorial we will be taking a look at working
with functions.
We will start off by looking for the domain and range. Then, we
will
move on to finding functional values. Don't be thrown by the
different
way function notation looks. When you go to evaluate these
functions,
we are just plugging in values for x and
simplifying,
just like you have done a million times before (it
is one of those types of problems that I warn students to not make
harder
than it is). Just think
of finding a functional value as a fancy assignment statement. I
think we are ready to forge ahead.

Tutorial
A relation is a set of ordered pairs where the
first components
of the ordered pairs are the input values and the second components are
the output values. 
A function is a relation that assigns to each
input number EXACTLY
ONE output number.
Be careful. Not every
relation is a function.
A function has to fit the above definition to a tee. 
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. 
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. 
Example
1: Find the domain and range of the relation.
Also,
determine whether the relation is a function. {(3, 2), (4, 3), (5, 4),
(6, 5)} 
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. So,
what do you get for the domain?
If you got {3, 4, 5, 6}, you are correct!
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. SO,
what do you get for the range?
If you got {2, 3, 4, 5}, you are absolutely right!
Is this a
function or not?
We need to ask ourselves, does every
first element
(or input) correspond with EXACTLY ONE second element (or
output)?
In this case, the answer is yes. 3 only goes with 2, 4 only goes
with 3, 5 only goes with 4 and 6 only goes with 5.
So, this relation would be an
example of a
function. 
Example
2: Find the domain and range of the relation.
Also,
determine whether the relation is a function.

Reading the ordered pairs off of the graph, we
get
{(2, 3), (2, 4), (3, 3), (4, 3)}
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. So,
what do you get for the domain?
If you got {2, 3, 4}, you are correct!
Note that if any value repeats, we only need to list it
one time.
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. So,
what do you get for the range?
If you got {3, 4}, you are absolutely right!
Again, note that if any value repeats, we only need to
list it one time.
Is this a
function or not?
We need to ask ourselves, does every first element (or input)
correspond
with exactly one second element (or output)? In this case, the
answer
is NO. The input value of 2 goes with two output values, 3 and
4.
It only takes one input value to associate with more than one output
value
to be invalid as a function.
In this case, the relation is not a function. 
Example
3: Find the domain and range of the relation.
Also,
determine whether the relation is a function.

Reading the ordered pairs off of the diagram we
get
{(a, 1), (b, 2), (c, 1), (d,
2)}
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. So,
what do you get for the domain?
If you got {a, b, c, d},
you are correct!
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. So,
what do you get for the range?
If you got {1, 2}, you are absolutely right!
Is this a
function or not?
We need to ask ourselves, does every
first element
(or input) correspond with exactly one second element (or
output)?
In this case, the answer is yes. a only
goes with 1, b only goes with 1, c only
goes with 2 and d only goes with 2.
Note that a relation can still
be a function
if an output value associates with more than one input value as shown
in
this example. But again, it would be a no no the other way
around,
where an input value corresponds to two or more output values.
So, this relation would be an
example of a
function. 
Example
4: Decide whether y is a function of x. 
To check if y is a
function of x,
we need to solve for y first and then
check
to see if there is only one output for every input. 

*Inverse of add x is
sub. x
*Inverse of mult. by 5 is div.
by 5

At this point we ask ourselves, would we get one value
for y if you plug in any value for x?
If you answered yes, you are right on. For example, if
we plugged in
a 1 for x, then y would only equal one value, 2/5.
Note that since it is solved for y, y is
our output value and x is our input
value.
Since our answer to that question is yes, that means
by definition, y is a function of x. 
Example
5: Decide whether y is a function of x. 
To check if y is a
function of x,
we need to solve for y squared first
and then
check to see if there is only one output for every input. 

*Inverse of add x is sub. x

At this point we ask ourselves, would we get one value
for y if you plug in any value for x?
If you answered no, you are correct.
For example, if our input value x is 16,
then our output value y could either be 4 or  4. Note
that
I could have picked an infinite number of examples like this one.
You only need to show one example where the input value is associated
with
more than one output value to disqualify it from being a function.
This means that at least one input value is
associated with more
than one output value, so by definition, y is not a function of x. 
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
6: Find the domain and range of each
relation.
Use the vertical line test to determine whether each graph is a graph
of
a function.

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 the
line
has arrows at both ends, that means it would go on and on forever
to
the right and to the left.
This means that the domain is {x  x is a real number}.
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 line has
arrows
at both ends, that means it would go on and on forever up and
down.
This means that the range is {y  y is a real number}.
Vertical Line
Test
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
7: Find the domain and range of each
relation.
Use the vertical line test to determine whether each graph is a graph
of
a function.

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 that the
farthest left point is (1, 0) and the farthest right point is (5, 0),
and
the circle is enclosed between these values. That means that if
we
wrote out ordered pairs for all the values going around the circle, we
would only use the values from 1 to 5 for x.
This means that the domain is {x  1 <x< 5}.
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 that the
highest
point is (3, 2) and the lowest point is (3, 2), and the circle is
enclosed
between these values. That means that if we wrote out ordered
pairs
for all the values going around the circle, we would only use the
values
from 2 to 2 for y.
This means that the range is {y 
2 < y < 2}
Vertical Line
Test
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: (3, 2) and (3, 2). This shows that
the input value of 3 associates with two output values, which is not
acceptable
in the function world.
Therefore, this is not a graph of a function. 
Function Notation
f(x) read “f of x”

Example
8: Find the functional values f(0), f(1), and
f(1)
for the function . 
Again, think of functional notation as a fancy
assignment statement.
For example, when we are looking for f(0),
we are going to plug in 0 for the value of x in
our function f and so forth. 

*Plug in 0 for x and evaluate
*Plug in 1 for x and evaluate
*Plug in 1 for x and evaluate

So our answers are f(0)
= 1, f(1)
= 4, and f(1) = 2. 
Example
9: Find the functional values g(1), g(1/2), and g(5)
for the function . 

*Plug in 1 for x and evaluate
*Plug in 1/2 for x and evaluate
*Plug in 5 for x and evaluate

So our answers are g(1)
= 4, g(1/2)
= 1, and g(5) = 46. 
Example
10: Find the functional values h(1), h(1/2),
and
h(2) for the function h(x) = 3. 
This is what is called a constant function. That means,
no matter what the input value is, this functional value is ALWAYS
going
to be 3.
Note how there is no input value visible. This is
not the same
as having 3x. 

*This is a constant function that
is always
3
*This is a constant function
that is always
3
*This is a constant function
that is always
3 
So our answers are h(1)
= 3, h(1/2)
= 3, and h(2) = 3. 
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
Problem 1a  1b: Find the domain and range of the
relation. Also,
determine whether the relation is a function.
Practice
Problems 2a  2b: Find the domain and range of each
relation. Use
the vertical line test to determine whether each graph is a graph of a
function.
Practice
Problems 3a  3b: Decide whether y is a
function of x.
Practice
Problems 4a  4b: Find the functional values.
Need Extra Help on these Topics?
Last revised on July 3, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.
