Objectives
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Determine if a relation, equation, or graph defines a function using the definition as well as the vertical line test.
Section 2.1 Introduction to Functions (FN1)
Subsection 2.1.1 Activities
Definition 2.1.1.
A relation is a relationship between sets of values. Relations in mathematics are usually represented as ordered pairs: (input, output) or \((x, y)\text{.}\) When observing relations, we often refer to the \(x\)-values as the domain and the \(y\)-values as the range.
Definition 2.1.2.
Mapping Notation (also known as an arrow diagram) is a way to show relationships visually between sets. For example, suppose you are given the following ordered pairs: \((3, -8), (4,6)\text{,}\) and \((2,-1)\text{.}\) Each of the \(x\)-values "map onto" a \(y\)-value and can be visualized in the following way:
Diagram Exploration Keyboard Controls
Notice that an arrow is used to indicate which \(x\)-value is mapped onto its corresponding \(y\)-value.
Activity 2.1.4.
Use mapping notation to create a visual representation of the following relation.
\begin{equation*}
(-1,5), (2,6), (4,-2)
\end{equation*}
(a)
(b)
Activity 2.1.5.
Use mapping notation to create a visual representation of the following relation.
\begin{equation*}
(6,4), (3,4), (6,5)
\end{equation*}
(a)
(b)
Activity 2.1.6.
Use mapping notation to create a visual representation of the following relation.
\begin{equation*}
(1,2), (-5,2), (-7,2)
\end{equation*}
(a)
(b)
Activity 2.1.7.
Use mapping notation to create a visual representation of the following relation.
\begin{equation*}
(3,-2), (-4,-1), (3,5)
\end{equation*}
(a)
(b)
Activity 2.1.8.
Now that you have created a visual representation for each of Activity 2.1.4, Activity 2.1.5, Activity 2.1.6, and Activity 2.1.7, what similarities do you see? What differences?
Activity 2.1.9.
(a)
(b)
(c)
(d)
Definition 2.1.10.
A function is a relation where every input (or \(x\)-value) is mapped onto exactly one output (or \(y\)-value).
Note that all functions are relations but not all relations are functions!
Remark 2.1.11.
We see that in Activity 2.1.9, the McRonald’s menu is a function. There is no confusion in determining the cost of a given menu item. However, Burger Queen’s menu is not a function. There is a discrepancy in the cost of nuggets, and the menu could cause confusion.
Activity 2.1.12.
Relations can be expressed in multiple ways (ordered pairs, tables, and verbal descriptions).
(a)
Let’s revisit some of the sets of ordered pairs we’ve previously explored in Activity 2.1.4, Activity 2.1.5, and Activity 2.1.6. Which of the following sets of ordered pairs represent a function?
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\(\displaystyle (-1,5), (2,6), (4,-2)\)
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\(\displaystyle (6,4), (3,4), (6,5)\)
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\(\displaystyle (1,2), (-5,2), (-7,2)\)
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\(\displaystyle (-1,2), (-1,9), (1,9)\)
(b)
Note that relations can be expressed in a table. A table of values is shown below. Is this an example of a function? Why or why not?
| \(x\) | \(y\) |
|---|---|
| \(-5\) | \(-2\) |
| \(-4\) | \(-5\) |
| \(-2\) | \(8\) |
| \(8\) | \(-4\) |
| \(8\) | \(1\) |
(c)
Relations can also be expressed in words. Suppose you are looking at the amount of time you spend studying versus the grade you earn in your Algebra class. Is this an example of a function? Why or why not?
Remark 2.1.13.
Notice that when trying to determine if a relation is a function, we often have to rely on looking at the domain and range values. Thus, it is important to be able to idenfity the domain and range of any relation!
Activity 2.1.14.
For each of the given functions, determine the domain and range.
(a)
\((-4,3), (-1,8), (7,4), (1,9)\)
(b)
Diagram Exploration Keyboard Controls
(c)
| \(x\) | \(y\) |
|---|---|
| \(-2\) | \(5\) |
| \(0\) | \(4\) |
| \(3\) | \(6\) |
| \(8\) | \(1\) |
(d)
The amount of time you spend studying versus the grade you earn in your Algebra class.
Activity 2.1.15.
Determine whether each of the following relations is a function.
(a)
Diagram Exploration Keyboard Controls
(b)
Diagram Exploration Keyboard Controls
(c)
Diagram Exploration Keyboard Controls
Remark 2.1.16.
You probably noticed (in Activity 2.1.15) that when the graph has points that "line up" or are on top of each other, they have the same \(x\)-values. When this occurs, this shows that the same \(x\)-value has two different outputs (\(y\)-values) and that the relation is not a function.
Definition 2.1.17.
The vertical line test is a method used to determine whether a relation on a graph is a function.
Start by drawing a vertical line anywhere on the graph and observe the number of times the relation on the graph intersects with the vertical line. If every possible vertical line intersects the graph at only one point, then the relation is a function. If, however, the graph of the relation intersects a vertical line more than once (anywhere on the graph), then the relation is not a function.
Activity 2.1.18.
Use the vertical line test (Definition 2.1.17) to determine whether each graph of a relation represents a function.
(a)
Diagram Exploration Keyboard Controls
(b)
Diagram Exploration Keyboard Controls
(c)
Diagram Exploration Keyboard Controls
(d)
Diagram Exploration Keyboard Controls
Activity 2.1.19.
Let’s explore how to determine whether an equation represents a function.
(a)
Suppose you are given the equation \(x=y^2\text{.}\)
-
If \(x=4\text{,}\) what kind of \(y\)-values would you get for \(x=y^2\text{?}\)
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Based on this information, do you think \(x=y^2\) is a function?
(b)
Suppose you are given the equation \(y=3x^2+2\text{.}\)
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If \(x=4\text{,}\) what kind of \(y\)-values would you get for \(y=3x^2+2\text{?}\)
-
Based on this information, do you think \(y=3x^2+2\) is a function?
(c)
Suppose you are given the equation \(x^2+y^2=25\text{.}\)
-
If \(x=4\text{,}\) what kind of \(y\)-values would you get for \(x^2+y^2=25\text{?}\)
-
Based on this information, do you think \(x^2+y^2=25\) is a function?
(d)
Suppose you are given the equation \(y=-4x-3\text{.}\)
-
If \(x=4\text{,}\) what kind of \(y\)-values would you get for \(y=-4x-3\text{?}\)
-
Based on this information, do you think \(y=-4x-3\) is a function?
(e)
How can you look at an equation to determine whether or not it is a function?
Remark 2.1.20.
Notice that Activity 2.1.19 shows that equations with a \(y^2\) term generally do not define functions. This is because to solve for a squared variable, you must consider both positive and negative inputs. For example, both \(2^2=4\) and \((-2)^2=4\text{.}\)
Activity 2.1.21.
It’s important to be able to determine the domain of any equation, especially when thinking about functions. Answer the following questions given the equation \(y=\sqrt{x-2}\text{.}\)
(a)
Which of the following values of \(x\) would cause \(y\) to be undefined (if any)?
-
\(\displaystyle -2\)
-
\(\displaystyle 0\)
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\(\displaystyle 2\)
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\(\displaystyle 4\)
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none of the above
(b)
Based on this information, for which of the following values of \(x\) would \(y\) be defined?
-
\(\displaystyle -2\)
-
\(\displaystyle 0\)
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\(\displaystyle 2\)
-
\(\displaystyle 4\)
-
none of the above
(c)
There are many more \(x\)- values than just those found above that, when plugged in, give a defined value for \(y\text{.}\) How can we represent the domain of this equation in interval notation?
-
\(\displaystyle (-\infty,2)\)
-
\(\displaystyle (-\infty,2]\)
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\(\displaystyle [2,\infty)\)
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\(\displaystyle (2,\infty)\)
-
\(\displaystyle (-\infty,\infty)\)
Activity 2.1.22.
Answer the following questions given the equation \(y=-5x+1\text{.}\)
(a)
Which of the following values of \(x\) would cause \(y\) to be undefined (if any)?
-
\(\displaystyle -2\)
-
\(\displaystyle 0\)
-
\(\displaystyle 4\)
-
\(\displaystyle -5\)
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none of the above
(b)
Based on this information, for which of the following values of \(x\) would \(y\) be defined?
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\(\displaystyle -2\)
-
\(\displaystyle 0\)
-
\(\displaystyle 4\)
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\(\displaystyle -5\)
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none of the above
(c)
How can we represent the domain of this equation in interval notation?
-
\(\displaystyle (-\infty,0)\)
-
\(\displaystyle (0,\infty)\)
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\(\displaystyle (-5, 1)\)
-
\(\displaystyle (-\infty,\infty)\)
Activity 2.1.23.
Answer the following questions given the equation \(y=\dfrac{3}{x-5}\text{.}\)
(a)
Which of the following values of \(x\) would cause \(y\) to be undefined (if any)?
-
\(\displaystyle -3\)
-
\(\displaystyle 0\)
-
\(\displaystyle -4\)
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\(\displaystyle 5\)
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none of the above
(b)
Based on this information, for which of the following values of \(x\) would \(y\) be defined?
-
\(\displaystyle -3\)
-
\(\displaystyle 0\)
-
\(\displaystyle -4\)
-
\(\displaystyle 5\)
-
none of the above
(c)
How can we represent the domain of this equation in interval notation?
-
\(\displaystyle (-\infty,5)\)
-
\(\displaystyle (5,\infty)\)
-
\(\displaystyle (-5, 5)\)
-
\((-\infty,5)\)U\((5,\infty)\)
Activity 2.1.24.
Find the domain of each of the following functions. Write your answer in interval notation.
(a)
\(f(x)=\dfrac{x+3}{(x-2)(x+5)}\)
(b)
\(f(x)=\sqrt{2x-5}\)
(c)
\(f(x)=\dfrac{2}{\sqrt{4-x}}\)
Remark 2.1.25.
When determining the domain of an equation, it is often easier to first find values of \(x\) that make the function undefined. Once you have those values, then you know that \(x\) can be any value but those.
Subsection 2.1.2 Exercises
Exercises available at
https://tbil.org/preview/precalculus/exercises/#/bank/FN1/
