ORCCA Domain and Range

Section 10.2 Domain and Range

A function is a process for turning input values into output values. Occasionally a function $f$ will have input values for which the process breaks down.

Subsection 10.2.1 Domain

Example 10.2.1.

Let $P$ be the population of Portland as a function of the year. According to Google¹ we can say that:

\begin{align*} P(2016) \amp = 639863 \amp P(1990) \amp= 487849 \end{align*}

But what if we asked to find $P(1600)\text{?}$ The question doesn't really make sense anymore. The Multnomah tribe lived in villages in the area, but the city of Portland was not incorporated until 1851. We say that $P(1600)$ is undefined.

https://www.google.com/publicdata/explore?ds=kf7tgg1uo9ude_&met_y=population&hl=en&dl=en#!ctype=l&strail=false &bcs=d&nselm=h&met_y=population&scale_y=lin&ind_y=false&rdim=country&idim=place:4159000&ifdim=country &hl=en_US&dl=en&ind=false

Example 10.2.2.

If $m$ is a person's mass in kg, let $w(m)$ be their weight in lb. There is an approximate formula for $w\text{:}$

\begin{equation*} w(m)\approx 2.2m \end{equation*}

From this formula we can find:

\begin{align*} w(50) \amp \approx 110 \amp w(80) \amp \approx 176\text{,} \end{align*}

which tells us that a $50$- kg person weighs 110 lb, and an $80$- kg person weighs 176 lb.

What if we asked for $w(-100)\text{?}$ In the context of this example, we would be asking for the weight of a person whose mass is -100 kg. This is clearly nonsense. That means that $w(-100)$ is undefined. Note that the context of the example is telling us that $w(-100)$ is undefined even though the formula alone might suggest that $w(-100)=-220\text{.}$

Example 10.2.3.

Let $g$ have the formula

\begin{equation*} g(x)=\frac{x}{x-7}\text{.} \end{equation*}

For most $x$-values, $g(x)$ is perfectly computable:

\begin{align*} g(2) \amp = -\frac{2}{5} \amp g(14) \amp = 2\text{.} \end{align*}

But if we try to compute $g(7)\text{,}$ we run into an issue of arithmetic.

\begin{align*} g(7) \amp = \frac{7}{7-7}\\ \amp = \frac{7}{0} \end{align*}

The expression $\frac{7}{0}$ is undefined. There is no number that this could equal.

Checkpoint 10.2.4.

These examples should motivate the following definition.

Definition 10.2.5. Domain.

The domain of a function $f$ is the collection of all of its valid input values.

Example 10.2.6.

Referring to the functions from Examples 10.2.1–10.2.3

The domain of $P$ is all years starting from 1851 and later. It would also be reasonable to say that the domain is actually all years from 1851 up to the current year, since we cannot guarantee that Portland will exist forever.
The domain of $w$ is all positive real numbers. It is nonsensical to have a person with negative mass or even one with zero mass. While there is some lower bound for the smallest mass a person could have, and also an upper bound for the largest mass a person could have, these boundaries are gray. We can say for sure that non-positive numbers should never be used as inputs for $w\text{.}$
The domain of $g$ is all real numbers except $7\text{.}$ This is the only number that causes a breakdown in $g$'s formula.

Subsection 10.2.2 Interval, Set, and Set-Builder Notation

Communicating the domain of a function can be wordy. In mathematics, we can communicate the same information using concise notation that is accepted for use almost everywhere. Figure 10.2.7 contains example functions from this section and their domains, and demonstrates interval notation for these domains. Basic interval notation is covered in Section 1.7, but some of our examples here go beyond what that section covers.

a number line with a mark at 0; there is a parenthesis at 0 that opens to the right and a thick line overlays the number line and points to the right — Figure 10.2.7. Domains from Earlier Examples

Again, basic interval notation is covered in Section 1.7, but one thing appears in Figure 10.2.7 that is not explained in that earlier section: the $\cup$ symbol, which we see in the domain of $g\text{.}$

Occasionally there is a need to consider number line pictures such as Figure 10.2.8, where two or more intervals appear.

a number line with marks at -5, 1, 4, and 7; there is a pair of brackets at -5 and 0 with a thick line between them. There is a parenthesis opening to the right at 4 and a bracket opening to the left at 7 with a thick line between them — Figure 10.2.8. A number line with a union of two intervals

This picture is trying to tell you to consider numbers that are between $-5$ and $1\text{,}$ together with numbers that are between $4$ and $7\text{.}$ That word “together” is related to the word “union,” and in math the union symbol, $\cup\text{,}$ captures this idea. So we can write the numbers in this picture as

\begin{equation*} [-5,1]\cup(4,7] \end{equation*}

(which uses interval notation).

With the domain of $g$ in Figure 10.2.7, the number line picture shows us another “union” of two intervals. They are very close together, but there are still two separated intervals in that picture: $(-\infty,7)$ and $(7,\infty)\text{.}$ Their union is represented by $(-\infty,7)\cup(7,\infty)\text{.}$

Checkpoint 10.2.9.

Checkpoint 10.2.10.

Sometimes we will consider collections of only a short list of numbers. In those cases, we use set notation (first introduced in Section 1.6). With set notation, we have a list of numbers in mind, and we simply list all of those numbers. Curly braces are standard for surrounding the list. Figure 10.2.11 illustrates set notation in use.

a number line with marks at -2 and 3 — Figure 10.2.11. Set Notation

a number line with marks at -5, 1, 3 and 5 — Figure 10.2.11. Set Notation

Checkpoint 10.2.12.

While most collections of numbers that we will encounter can be described using a combination of interval notation and set notation, there is another commonly used notation that is very useful in algebra: set-builder notation, which was introduced in Section 1.7. Set-builder notation also uses curly braces. Set-builder notation provides a template for what a number that is under consideration might look like, and then it gives you restrictions on how to use that template. A very basic example of set-builder notation is

\begin{equation*} \{x\mid x\geq3\}\text{.} \end{equation*}

Verbally, this is “the set of all $x$ such that $x$ is greater than or equal to $3\text{.}$” Figure 10.2.13 gives more examples of set-builder notation in use.

a number line with marks at -2 and 3; there is a parenthesis that opens to the right at -2 and a bracket that opens to the left at 3; there is a thick line in between them — Figure 10.2.13. Set-Builder Notation

Checkpoint 10.2.14.

Example 10.2.15.

What is the domain of the function $A\text{,}$ where $A(x)=\frac{2x+1}{x^2-2x-8}\text{?}$

Note that if you plugged in some value for $x\text{,}$ the only thing that might go wrong is if the denominator equals $0\text{.}$ So a bad value for $x$ would be when

\begin{align*} x^2 - 2x - 8\amp=0\\ (x + 2)(x - 4)\amp=0 \end{align*}

Here, we used a basic factoring technique from Section 7.3. To continue, either

\begin{align*} x + 2\amp=0\amp\text{or}\amp\amp x-4\amp=0\\ x\amp=-2\amp\text{or}\amp\amp x\amp=4\text{.} \end{align*}

These are the bad $x$-values because they lead to division by $0$ in the formula for $A\text{.}$ So on a number line, if we wanted to picture the domain of $A\text{,}$ we would make a sketch like:

So the domain is the union of three intervals: $(-\infty,-2)\cup(-2,4)\cup(4,\infty)\text{.}$

Example 10.2.16.

What is the domain of the function $B\text{,}$ where $B(x)=\sqrt{7-x}+3\text{?}$

Note that if you plugged in some value for $x\text{,}$ the only thing that might go wrong is if the value in the radical is negative. So the good values for $x$ would be when

\begin{align*} 7 - x\amp\geq0\\ 7\geq x\\ x\leq 7 \end{align*}

So on a number line, if we wanted to picture the domain of $B\text{,}$ we would make a sketch like:

So the domain is the interval $(-\infty,7]\text{.}$

There are three main properties of algebraic functions that cause numbers to be excluded from a domain, which are summarized here.

List 10.2.17. Summary of Algebraic Domain Restrictions

Denominators

Division by zero is undefined. So if a function contains an expression in a denominator, it will only be defined where that expression is not equal to zero.

Example 10.2.15 demonstrates this.

Square Roots

The square root of a negative number is undefined. So if a function contains a square root, it will only be defined when the expression inside that radical is greater than or equal to zero. (This is actually true for any even $n$th radical.)

Example 10.2.16 demonstrates this.

Context

Some numbers are nonsensical in context. If a function has real-world context, then this may add additional restrictions on the input values.

Example 10.2.2 demonstrates this.

Subsection 10.2.3 Range

The domain of a function is the collection of its valid inputs; there is a similar notion for output.

Definition 10.2.18. Range.

The range of a function $f$ is the collection of all of its possible output values.

Example 10.2.19.

Let $f$ be the function defined by the formula $f(x)=x^2\text{.}$ Finding $f$'s domain is straightforward. Any number anywhere can be squared to produce an output, so $f$ has domain $(-\infty,\infty)\text{.}$ What is the range of $f\text{?}$

Explanation

We would like to describe the collection of possible numbers that $f$ can give as output. We will use a graphical approach. Figure 10.2.20 displays a graph of $f\text{,}$ and the visualization that reveals $f$'s range.

A parabola with vertex at (0,0) that opens upward — Figure 10.2.20. $y=f(x)$ where $f(x)=x^2\text{.}$ The second graph illustrates how to visualize the range. In the third graph, the range is marked as an interval on the $y$-axis.

The same parabola is pictured with arrows pointing from each point toward the y-axis. — Figure 10.2.20. $y=f(x)$ where $f(x)=x^2\text{.}$ The second graph illustrates how to visualize the range. In the third graph, the range is marked as an interval on the $y$-axis.

Output values are the $y$-coordinates in a graph. If we “slide the ink” left and right over to the $y$-axis to emphasize what the $y$-values in the graph are, we have $y$-values that start from $0$ and continue upward forever. Therefore, the range is $[0,\infty)\text{.}$

Warning 10.2.21. Finding range from a formula.

Sometimes it is possible to compute a range without the aid of a graph. However, doing so can often require techniques covered in calculus. Therefore, when you are asked to find the range of a function based on its formula, your approach will most often need to be a graphical one.

Example 10.2.22.

Given the function $g$ graphed in Figure 10.2.23, find the domain and range of $g\text{.}$

Figure 10.2.23. $y=g(x)$

To find the domain, we can visualize all of the $x$-values that are valid inputs for this function by “sliding the ink” down onto the $x$-axis. The arrows at the far left and far right of the curve indicate that whatever pattern we see in the graph continues off to the left and right. Here, we see that the arms of the graph appear to be tapering down to the $x$-axis and extending left and right forever. Every $x$-value can be used to get an output for the function, so the domain is $(-\infty,\infty)\text{.}$

If we visualize the possible outputs by “sliding the ink” sideways onto the $y$-axis, we find that outputs as high as $3$ are possible (including $3$ itself). The outputs appear to get very close to $0$ when $x$ is large, but they aren't quite equal to $0\text{.}$ So the range is $(0,3]\text{.}$

The same bell-shaped graph with the x-axis highlighted and arrows pointing upward to the points on the graph — Figure 10.2.24. Domain of $g$

The same bell-shaped graph with arrows going from points on the graph to the y-values on the y-axis; there is an interval on the y-axis indicating the range — Figure 10.2.25. Range of $g$

Checkpoint 10.2.26.

The examples of finding domain and range so far have all involved either a verbal description of a function, a formula for that function, or a graph of that function. Recall that there is a fourth perspective on functions: a table. In the case of a table, we have very limited information about the function's inputs and outputs. If the table is all that we have, then there are a handful of input values listed in the table for which we know outputs. For any other input, the output is undefined.

Example 10.2.27.

Consider the function $k$ given in Table 10.2.28. What is the domain and range of $k\text{?}$

Table 10.2.28.

$x$	$k(x)$
$3$	$4$
$8$	$5$
$10$	$5$

Explanation

All that we know about $k$ is that $k(3)=4\text{,}$ $k(8)=5\text{,}$ and $k(10)=5\text{.}$ Without any other information such as a formula for $k$ or a context for $k$ that tells us its verbal description, we must assume that its domain is $\{3,8,10\}\text{;}$ these are the only valid input for $k\text{.}$ Similarly, $k$'s range is $\{4,5\}\text{.}$

Note that we have used set notation, not interval notation, since the answers here were lists of $x$-values (for the domain) and $y$-values (for the range). Also note that we could graph the information that we have about $k$ in Figure 10.2.29, and the visualization of “sliding ink” to determine domain and range still works.

Figure 10.2.29.

Exercises 10.2.4 Exercises

Review and Warmup

1.

For the interval expressed in the number line, write it using set-builder notation and interval notation.

In set-builder notation:

In interval notation:

2.

For the interval expressed in the number line, write it using set-builder notation and interval notation.

In set-builder notation:

In interval notation:

3.

For the interval expressed in the number line, write it using set-builder notation and interval notation.

In set-builder notation:

In interval notation:

4.

For the interval expressed in the number line, write it using set-builder notation and interval notation.

In set-builder notation:

In interval notation:

Domain and Range From a Graph

5.

A function is graphed.

This function has domain and range .

6.

A function is graphed.

This function has domain and range .

7.

A function is graphed.

This function has domain and range .

8.

A function is graphed.

This function has domain and range .

9.

A function is graphed.

This function has domain and range .

10.

A function is graphed.

This function has domain and range .

11.

A function is graphed.

This function has domain and range .

12.

A function is graphed.

This function has domain and range .

13.

A function is graphed.

This function has domain and range .

14.

A function is graphed.

This function has domain and range .

15.

A function is graphed.

This function has domain and range .

16.

A function is graphed.

This function has domain and range .