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Precalculus for Team-Based Inquiry Learning 2025 Edition — Instructor Version

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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.
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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:
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Figure 2.1.3. Every \(x\)-value from the ordered pair list is listed in the input set and every \(y\)-value is listed in the output set. An arrow is drawn from every \(x\)-value to its corresponding \(y\)-value.
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Notice that an arrow is used to indicate which \(x\)-value is mapped onto its corresponding \(y\)-value.
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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*}
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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*}
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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*}
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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*}
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Activity 2.1.9.

Consider the menus shown below for two different fast food restaurants. We’ll consider the Items as inputs (\(x\)-values) and the Cost as outputs (\(y\)-values).
McRonald’s Item Cost
Nuggets \(\$5\)
Burger \(\$6\)
Fries \(\$3\)
Nachos \(\$6\)
Burger Queen’s Item Cost
Nuggets \(\$4\)
Burger \(\$8\)
Fries \(\$2\)
Nuggets \(\$7\)
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(a)
How much would it cost to get nuggets and fries at McRonald’s?
  1. \(\displaystyle \$8\)
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  2. \(\displaystyle \$9\)
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  3. \(\displaystyle \$11\)
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  4. \(\displaystyle \$12\)
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Answer.
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(b)
How much would it cost to get nuggets and fries at Burger Queen?
  1. \(\displaystyle \$6\)
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  2. \(\displaystyle \$9\)
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  3. \(\displaystyle \$10\)
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  4. \(\displaystyle \$12\)
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Answer.
A, B, or not able to be determined. Notice that that menu is not a function since nuggets are a repeated item.
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(c)
Notice that burgers and nachos cost the same amount at McRonald’s. Is that an issue? Explain your reasoning.
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Answer.
No. It’s okay for two items to have the same price.
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(d)
Notice on the Burger Queen menu that nuggets cost \(\$4\) in one spot but \(\$7\) in another spot. Is that an issue? Explain your reasoning.
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Answer.
Yes. It’s not clear how much nuggets cost.
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Definition 2.1.10.

A function is a relation where every input (or \(x\)-value) is mapped onto exactly one output (or \(y\)-value).
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Note that all functions are relations but not all relations are functions!
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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.
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Activity 2.1.12.

Relations can be expressed in multiple ways (ordered pairs, tables, and verbal descriptions).
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(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?
  1. \(\displaystyle (-1,5), (2,6), (4,-2)\)
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  2. \(\displaystyle (6,4), (3,4), (6,5)\)
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  3. \(\displaystyle (1,2), (-5,2), (-7,2)\)
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  4. \(\displaystyle (-1,2), (-1,9), (1,9)\)
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Answer.
A and C
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(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?
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\(x\) \(y\)
\(-5\) \(-2\)
\(-4\) \(-5\)
\(-2\) \(8\)
\(8\) \(-4\)
\(8\) \(1\)
Answer.
This relation is not a function. The \(x\)-value, \(8\text{,}\) has two outputs.
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(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?
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Answer.
Yes, this is an example of a function.
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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!
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Activity 2.1.14.

For each of the given functions, determine the domain and range.
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(a)
\((-4,3), (-1,8), (7,4), (1,9)\)
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Answer.
Domain: {\(-4, -1, 1, 7\)}; Range: {\(3, 4, 8, 9\)}
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(c)
\(x\) \(y\)
\(-2\) \(5\)
\(0\) \(4\)
\(3\) \(6\)
\(8\) \(1\)
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Answer.
Domain: {\(-2, 0, 3, 8\)}; Range: {\(1, 4, 5, 6\)}
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(d)
The amount of time you spend studying versus the grade you earn in your Algebra class.
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Answer.
Domain: {time studying}; Range: {grade in Algebra class}
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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.
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Definition 2.1.17.

The vertical line test is a method used to determine whether a relation on a graph is a function.
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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.
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Activity 2.1.18.

Use the vertical line test (Definition 2.1.17) to determine whether each graph of a relation represents a function.
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(a)
Answer.
This graph does not pass the vertical line test and is therefore NOT a function.
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(b)
Answer.
This graph passes the vertical line test and is therefore a function.
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(c)
Answer.
This graph passes the vertical line test and is therefore a function.
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(d)
Answer.
This graph does not pass the vertical line test and is therefore NOT a function.
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Activity 2.1.19.

Let’s explore how to determine whether an equation represents a function.
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(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?
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Answer.
If \(x=4\text{,}\) \(y\) could be \(-2\) or \(2\text{.}\) Given that there are two outputs for a given \(x\text{,}\) then \(x=y^2\) is not a function.
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(b)
Suppose you are given the equation \(y=3x^2+2\text{.}\)
  • If \(x=4\text{,}\) what kind of \(y\)-values would you get for \(y=3x^2+2\text{?}\)
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  • Based on this information, do you think \(y=3x^2+2\) is a function?
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Answer.
If \(x=4\text{,}\) \(y=50\text{.}\) For every input (\(x\)-value), there is exactly one output (\(y\)-value) so \(y=3x^2+2\) is a function.
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(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 \(x^2+y^2=25\text{?}\)
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  • Based on this information, do you think \(x^2+y^2=25\) is a function?
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Answer.
If \(x=4\text{,}\) \(y\) could be \(-3\) or \(3\text{.}\) Given that there are two outputs for a given \(x\text{,}\) then \(x^2+y^2=25\) is not a function.
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(d)
Suppose you are given the equation \(y=-4x-3\text{.}\)
  • If \(x=4\text{,}\) what kind of \(y\)-values would you get for \(y=-4x-3\text{?}\)
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  • Based on this information, do you think \(y=-4x-3\) is a function?
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Answer.
If \(x=4\text{,}\) \(y=-19\text{.}\) For every input (\(x\)-value), there is exactly one output (\(y\)-value) so \(y=-4x-3\) is a function.
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(e)
How can you look at an equation to determine whether or not it is a function?
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Answer.
Students may have different answers. Ideally, you want students to see that they can check whether a given relation is a function in more than one way. They can test values (as this activity did by leading them to plug in values) or by using the vertical line test on a graph.
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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 \((-2)^2=4\text{.}\)
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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 \(y=\sqrt{x-2}\text{.}\)
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(a)
Which of the following values of \(x\) would cause \(y\) to be undefined (if any)?
  1. \(\displaystyle -2\)
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  2. \(\displaystyle 0\)
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  3. \(\displaystyle 2\)
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  4. \(\displaystyle 4\)
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  5. none of the above
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Answer.
A and B
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(b)
Based on this information, for which of the following values of \(x\) would \(y\) be defined?
  1. \(\displaystyle -2\)
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  2. \(\displaystyle 0\)
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  3. \(\displaystyle 2\)
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  4. \(\displaystyle 4\)
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  5. none of the above
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Answer.
C and D
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(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?
  1. \(\displaystyle (-\infty,2)\)
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  2. \(\displaystyle (-\infty,2]\)
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  3. \(\displaystyle [2,\infty)\)
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  4. \(\displaystyle (2,\infty)\)
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  5. \(\displaystyle (-\infty,\infty)\)
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Answer.
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Activity 2.1.22.

Answer the following questions given the equation \(y=-5x+1\text{.}\)
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(a)
Which of the following values of \(x\) would cause \(y\) to be undefined (if any)?
  1. \(\displaystyle -2\)
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  2. \(\displaystyle 0\)
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  3. \(\displaystyle 4\)
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  4. \(\displaystyle -5\)
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  5. none of the above
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Answer.
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(b)
Based on this information, for which of the following values of \(x\) would \(y\) be defined?
  1. \(\displaystyle -2\)
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  2. \(\displaystyle 0\)
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  3. \(\displaystyle 4\)
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  4. \(\displaystyle -5\)
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  5. none of the above
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Answer.
A, B, C, and D
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(c)
How can we represent the domain of this equation in interval notation?
  1. \(\displaystyle (-\infty,0)\)
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  2. \(\displaystyle (0,\infty)\)
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  3. \(\displaystyle (-5, 1)\)
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  4. \(\displaystyle (-\infty,\infty)\)
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Answer.
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Activity 2.1.23.

Answer the following questions given the equation \(y=\dfrac{3}{x-5}\text{.}\)
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(a)
Which of the following values of \(x\) would cause \(y\) to be undefined (if any)?
  1. \(\displaystyle -3\)
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  2. \(\displaystyle 0\)
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  3. \(\displaystyle -4\)
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  4. \(\displaystyle 5\)
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  5. none of the above
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Answer.
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(b)
Based on this information, for which of the following values of \(x\) would \(y\) be defined?
  1. \(\displaystyle -3\)
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  2. \(\displaystyle 0\)
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  3. \(\displaystyle -4\)
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  4. \(\displaystyle 5\)
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  5. none of the above
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Answer.
A, B, and C
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(c)
How can we represent the domain of this equation in interval notation?
  1. \(\displaystyle (-\infty,5)\)
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  2. \(\displaystyle (5,\infty)\)
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  3. \(\displaystyle (-5, 5)\)
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  4. \((-\infty,5)\)U\((5,\infty)\)
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Answer.
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Activity 2.1.24.

Find the domain of each of the following functions. Write your answer in interval notation.
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(a)
\(f(x)=\dfrac{x+3}{(x-2)(x+5)}\)
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Answer.
\((-\infty,-5)\cup(-5,2)\cup(2,\infty)\)
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(b)
\(f(x)=\sqrt{2x-5}\)
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Answer.
\(\left[\dfrac{5}{2},\infty \right) \)
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(c)
\(f(x)=\dfrac{2}{\sqrt{4-x}}\)
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Hint.
Notice that this function has both a denominator and a root to consider!
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Answer.
\((-\infty,4)\)
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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.
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Subsection 2.1.2 Exercises

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