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

Relation

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.

Function

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.

Domain

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.

Range

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. example 2

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.

example 3

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.

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 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.

example 7

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}.

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”

f is the function name. Output values are also called functional values. Note that you can use any letter to represent a function name, f is a very common one used.

x is your input variable.

Think of functional notation as a fancy assignment statement. When you need to evaluate the function for a given value of x, you simply replace x with that given value and simplify. Just like you do when you are finding the value of an expression when you are given a number for your variable as shown in Tutorial 2: Algebraic Expressions. For example, if we are looking for f(0), we would plug in 0 for the value of x in our function f.

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

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.

1a. {(1, -1), (1, -2), (2, -3), (3, - 4)}
(answer/discussion to 1a)

1b.

(answer/discussion to 1b)

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.

2a.

problem 2a (answer/discussion to 2a)