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Domain and Range

Find the domain of a function defined by an equation.

Domain and Range

Find the domain and range of a function (IA 3.5.1)

A relation is any set of ordered pairs, (x,y). A special type of relation, called a function, is studied extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.

When studying functions, it’s important to be able to identify potential input values, called the domain, and potential output values, called the range.

Example

Try it.

Find the domain of the following function: {(2, 10),(3, 10),(4, 20),(5, 30),(6, 40)}.

Solution

First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.
D: {2,3,4,5,6}

Notice here we are using set notation to represent this collection of input values.

A graph of a function can always help in identifying domain and range. When graphing basic functions, we can scan the x-axis just as we read in English from left to right to help determine the domain. We will scan the y-axis from bottom to top to help determine the range. So, in finding both domain and range, we scan axes from smallest to largest to see which values are defined. Typically, we will use interval notation, where you show the endpoints of defined sets using parentheses (endpoint not included) or brackets (endpoint is included) to express both the domain, D, and range, R, of a relation or function.

Example

Try it.

Find the domain and range of the function f whose graph is shown below.

Solution

Scanning the x-axis from left to right helps us to see the graph is defined for x-values between –3 to 1, so the domain of f is (−3,1]. (Note that open points translate to use of parentheses in interval notation, while included points translate to use of brackets in interval notation.)

Scanning the y-values from the bottom to top of the graph helps us to see the graph is defined for y-values between 0 to –4, so the range is [−4,0].

When working with functions expressed as an equation, the following steps can help to identify the domain.

Condensed — the full section is in OpenStax College Algebra 2e.

Finding the Domain of a Function Defined by an Equation

In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.

We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See .

We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write $(0,\ 100].$ We will discuss interval notation in greater detail later.

Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.

Before we begin, let us review the conventions of interval notation:

The smallest number from the interval is written first.
The largest number in the interval is written second, following a comma.
Parentheses, ( or ), are used to signify that an endpoint value is not included, called exclusive.
Brackets, [ or ], are used to indicate that an endpoint value is included, called inclusive.

See for a summary of interval notation.

Condensed — the full section is in OpenStax College Algebra 2e.

Using Notations to Specify Domain and Range

In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example, $\{x|10\le x<30\}$ describes the behavior of $x$ in set-builder notation. The braces $\{\}$ are read as “the set of,” and the vertical bar | is read as “such that,” so we would read $\{x|10\le x<30\}$ as “the set of x-values such that 10 is less than or equal to $x,$ and $x$ is less than 30.”

compares inequality notation, set-builder notation, and interval notation.

To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol, $\cup ,$ to combine two unconnected intervals. For example, the union of the sets $\{2,3,5\}$ and $\{4,6\}$ is the set $\{2,3,4,5,6\}.$ It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

\[\{x|\ |x|\ge 3\}=(-\infty ,-3]\cup [3,\infty )\]

Condensed — the full section is in OpenStax College Algebra 2e.

Finding Domain and Range from Graphs

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See .

We can observe that the graph extends horizontally from $-5$ to the right without bound, so the domain is $[-5,\infty ).$ The vertical extent of the graph is all range values $5$ and below, so the range is $(-\infty ,5].$ Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.

Example

Try it.

Find the domain and range of the function $f$ whose graph is shown in .

Solution

We can observe that the horizontal extent of the graph is –3 to 1, so the domain of $f$ is $(-3,1].$

The vertical extent of the graph is 0 to –4, so the range is $[-4,0].$ See .

Example

Try it.

Find the domain and range of the function $f$ whose graph is shown in .

Solution

The input quantity along the horizontal axis is “years,” which we represent with the variable $t$ for time. The output quantity is “thousands of barrels of oil per day,” which we represent with the variable $b$ for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as $1973\le t\le 2008$ and the range as approximately $180\le b\le 2010.$

In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

Finding Domains and Ranges of the Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.

Example

Try it.

Find the domain and range of $f(x)=2{x}^{3}-x.$

Solution

There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

The domain is $(-\infty ,\infty )$ and the range is also $(-\infty ,\infty ).$

Example

Try it.

Find the domain and range of $f(x)=\frac{2}{x+1}.$

Solution

We cannot evaluate the function at $-1$ because division by zero is undefined. The domain is $(-\infty ,-1)\cup (-1,\infty ).$ Because the function is never zero, we exclude 0 from the range. The range is $(-\infty ,0)\cup (0,\infty ).$

Example

Try it.

Find the domain and range of $f(x)=2\sqrt{x+4}.$

Solution

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

\[x+4\ge 0\ \text{when }x\ge -4\]

The domain of $f(x)$ is $[-4,\infty ).$

We then find the range. We know that $f(-4)=0,$ and the function value increases as $x$ increases without any upper limit. We conclude that the range of $f$ is $[0,\infty ).$

Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function $f(x)=|x|.$ With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude, or modulus, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

If we input 0, or a positive value, the output is the same as the input.

\[f(x)=x\ \text{if}\ x\ge 0\]

If we input a negative value, the output is the opposite of the input.

\[f(x)=-x\ \text{if}\ x<0\]

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income $S$ would be $0.1S$ if $S\le \text{\$}10\text{,}000$ and $\text{\$}1000+0.2(S-\text{\$}10\text{,}000)$ if $S>\text{\$}10\text{,}000.$

Condensed — the full section is in OpenStax College Algebra 2e.

Key Concepts

The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.
The domain of a function can be determined by listing the input values of a set of ordered pairs. See .
The domain of a function can also be determined by identifying the input values of a function written as an equation. See , , and .
Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See .
For many functions, the domain and range can be determined from a graph. See and .
An understanding of toolkit functions can be used to find the domain and range of related functions. See , , and .
A piecewise function is described by more than one formula. See and .
A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See .

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Find the domain of the following function: {(2, 10),(3, 10),(4, 20),(5, 30),(6, 40)}.

விடை தெரியப்படுத்து

First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.
D: {2,3,4,5,6}
Notice here we are using set notation to represent this collection of input values.
Find the domain and range of the function f whose graph is shown below.

விடை தெரியப்படுத்து

Scanning the x-axis from left to right helps us to see the graph is defined for x-values between –3 to 1, so the domain of f is (−3,1]. (Note that open points translate to use of parentheses in interval notation, while included points translate to use of brackets in interval notation.)
Scanning the y-values from the bottom to top of the graph helps us to see the graph is defined for y-values between 0 to –4, so the range is [−4,0].
Find the domain of the function $f(x)={x}^{2}-3$

விடை தெரியப்படுத்து

The input value, shown by the variable x in the equation, is squared and then the result is lowered by three. Any real number may be squared and then be lowered by three, so there are no restrictions on the domain of this function. The domain is the set of real numbers.
In interval notation form, the domain of f is (−∞,∞).
For the relation {(1,3),(2,6),(3,9),(4,12),(5,15)}:
ⓐ Find the domain of the function, express using set notation.
ⓑ Find the range of the function, express using set notation.
Use the graph of the function to find its domain and range. Write the domain and range in interval notation.
D: _________ R: _________
Graph the following function below. Use this graph to help determine the domain and range and express using interval notation. $f(x)=-4x-3$
D: ________ R: _________
Use the graph of the function to find its domain and range. Write the domain and range in interval notation.
D: ________ R: _________
Graph the following function. Use this graph to help determine the domain and range and express using interval notation. $f(x)=-2|x|+3$
D: ________ R: _________
Graph the following function. Use this graph to help determine the domain and range and express using interval notation. $f(x)=\sqrt{x-3}$
D: _________ R: __________
Graph the following function. Use this graph to help determine the domain and range and express using interval notation. $f(x)=\sqrt[3]{x+4}$
D: _________ R: __________
Graph the following function. Use this graph to help determine the domain and range and express using interval notation. $f(x)=\frac{(x-3)}{(2x+1)}$
D: _________ R: __________
Find the domain of the following function: $\{(2,\ 10),(3,\ 10),(4,\ 20),(5,\ 30),(6,\ 40)\}$ .

விடை தெரியப்படுத்து

First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.
\[\{2,3,4,5,6\}\]
Find the domain of the function:
$\{(-5,4),(0,0),(5,-4),(10,-8),(15,-12)\}$

விடை தெரியப்படுத்து

$\{-5,\ 0,\ 5,\ 10,\ 15\}$
Find the domain of the function $f(x)={x}^{2}-1.$

விடை தெரியப்படுத்து

The input value, shown by the variable $x$ in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.
In interval form, the domain of $f$ is $(-\infty ,\infty ).$
Find the domain of the function: $f(x)=5-x+{x}^{3}.$

விடை தெரியப்படுத்து

$(-\infty ,\infty )$
Find the domain of the function $f(x)=\frac{x+1}{2-x}.$

விடை தெரியப்படுத்து

When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for $x.$
\[\begin{array}{lll}2-x & = & 0 \\ -x & = & -2 \\ x & = & 2\end{array}\]
Now, we will exclude 2 from the domain. The answers are all real numbers where $x<2$ or $x>2$ as shown in . We can use a symbol known as the union, $\cup ,$ to combine the two sets. In interval notation, we write the solution: $(-\infty ,2)\cup (2,\infty ).$
Find the domain of the function: $f(x)=\frac{1+4x}{2x-1}.$

விடை தெரியப்படுத்து

$(-\infty ,\frac{1}{2})\cup (\frac{1}{2},\infty )$
Find the domain of the function $f(x)=\sqrt{7-x}.$

விடை தெரியப்படுத்து

When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.
Set the radicand greater than or equal to zero and solve for $x.$
\[\begin{array}{lll}7-x & \ge & 0 \\ -x & \ge & -7 \\ x & \le & 7\end{array}\]
Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to $7,$ or $(-\infty ,7].$
Find the domain of the function $f(x)=\sqrt{5+2x}.$

விடை தெரியப்படுத்து

$[-\frac{5}{2},\infty )$
Describe the intervals of values shown in using inequality notation, set-builder notation, and interval notation.

விடை தெரியப்படுத்து

To describe the values, $x,$ included in the intervals shown, we would say, “ $x$ is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.”
Inequality $1\le x\le 3\ \text{or}\ x>5$
Set-builder notation $\{x|1\le x\le 3\ \text{or}\ x>5\}$
Interval notation $[1,3]\cup (5,\infty )$
Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.
Given , specify the graphed set in
1. ⓐ words
2. ⓑ set-builder notation
3. ⓒ interval notation
விடை தெரியப்படுத்து
1. ⓐ values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3
2. ⓑ $\{x|x\le -2\ \text{or}\ -1\le x<3\}$
3. ⓒ $(-\infty ,-2]\cup [-1,3)$
Find the domain and range of the function $f$ whose graph is shown in .

விடை தெரியப்படுத்து

We can observe that the horizontal extent of the graph is –3 to 1, so the domain of $f$ is $(-3,1].$
The vertical extent of the graph is 0 to –4, so the range is $[-4,0].$ See .
Find the domain and range of the function $f$ whose graph is shown in .

விடை தெரியப்படுத்து

The input quantity along the horizontal axis is “years,” which we represent with the variable $t$ for time. The output quantity is “thousands of barrels of oil per day,” which we represent with the variable $b$ for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as $1973\le t\le 2008$ and the range as approximately $180\le b\le 2010.$
In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.
Given , identify the domain and range using interval notation.

விடை தெரியப்படுத்து

domain =[1950,2002] range = [47,000,000,89,000,000]
Find the domain and range of $f(x)=2{x}^{3}-x.$

விடை தெரியப்படுத்து

There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.
The domain is $(-\infty ,\infty )$ and the range is also $(-\infty ,\infty ).$
Find the domain and range of $f(x)=\frac{2}{x+1}.$

விடை தெரியப்படுத்து

We cannot evaluate the function at $-1$ because division by zero is undefined. The domain is $(-\infty ,-1)\cup (-1,\infty ).$ Because the function is never zero, we exclude 0 from the range. The range is $(-\infty ,0)\cup (0,\infty ).$
Find the domain and range of $f(x)=2\sqrt{x+4}.$

விடை தெரியப்படுத்து

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.
\[x+4\ge 0\ \text{when }x\ge -4\]
The domain of $f(x)$ is $[-4,\infty ).$
We then find the range. We know that $f(-4)=0,$ and the function value increases as $x$ increases without any upper limit. We conclude that the range of $f$ is $[0,\infty ).$
Find the domain and range of $f(x)=-\sqrt{2-x}.$

விடை தெரியப்படுத்து

domain: $(-\infty ,2];$ range: $(-\infty ,0]$
A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, $n,$ to the cost, $C.$ Since one cannot have fractions of a person, this is really a discrete function. However, for this exercise we will treat it as a continuous function.

விடை தெரியப்படுத்து

Two different formulas will be needed. For n-values under 10, $C=5n.$ For values of $n$ that are 10 or greater, $C=50.$
\[C(n)=\{\begin{array}{lll}5n & \text{if} & 0
A cell phone company uses the function below to determine the cost, $C,$ in dollars for $g$ gigabytes of data transfer.
\[C(g)=\{\begin{array}{lll}25 & \text{if} & 0Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

விடை தெரியப்படுத்து

To find the cost of using 1.5 gigabytes of data, $C(1.5),$ we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.
\[C(1.5)=\text{\$}25\]
To find the cost of using 4 gigabytes of data, $C(4),$ we see that our input of 4 is greater than 2, so we use the second formula.
\[C(4)=25+10(4-2)=\text{\$}45\]
Sketch a graph of the function.
\[f(x)=\{\begin{array}{lll}{x}^{2} & \text{if} & x\le 1 \\ 3 & \text{if} & 12\end{array}\]

விடை தெரியப்படுத்து

Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality.
shows the three components of the piecewise function graphed on separate coordinate systems.
Now that we have sketched each piece individually, we combine them in the same coordinate plane. See .
Graph the following piecewise function.
\[f(x)=\{\begin{array}{lll}{x}^{3} & \text{if} & x<-1 \\ -2 & \text{if} & -14\end{array}\]
Why does the domain differ for different functions?

விடை தெரியப்படுத்து

The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.
How do we determine the domain of a function defined by an equation?
Explain why the domain of $f(x)=\sqrt[3]{x}$ is different from the domain of $f(x)=\sqrt[]{x}.$

விடை தெரியப்படுத்து

There is no restriction on $x$ for $f(x)=\sqrt[3]{x}$ because you can take the cube root of any real number. So the domain is all real numbers, $(-\infty ,\infty ).$ When dealing with the set of real numbers, you cannot take the square root of negative numbers. So $x$ -values are restricted for $f(x)=\sqrt[]{x}$ to nonnegative numbers and the domain is $[0,\infty ).$
When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?
How do you graph a piecewise function?

விடை தெரியப்படுத்து

Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the $x$ -axis and $y$ -axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate $-\infty$ or $\ \infty .$ Combine the graphs to find the graph of the piecewise function.
$f(x)=-2x(x-1)(x-2)$
$f(x)=5-2{x}^{2}$

விடை தெரியப்படுத்து

$(-\infty ,\infty )$
$f(x)=3\sqrt{x-2}$

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Domain and Range

Find the domain of a function defined by an equation.
Graph piecewise-defined functions.
Find the domain and range of a function (IA 3.5.1)
Identify the input values.
Identify any restrictions on the input and exclude those values from the domain.
Write the domain in interval notation form, if possible.
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Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

உங்களை முயற்சிக்கவும்

Parts of this page are adapted from OpenStax College Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

Inequality	\(1\le x\le 3\ \text{or}\ x>5\)
Set-builder notation	\(\{x\|1\le x\le 3\ \text{or}\ x>5\}\)
Interval notation	\([1,3]\cup (5,\infty )\)