2.2: Domain and Range
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 (Figure \(\PageIndex{2}\)).
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, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write \(\left(0, 100\right]\). 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 even 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 term from the interval is written first.
- The largest term in the interval is written second, following a comma.
- Parentheses, \((\) or \()\), are used to signify that an endpoint is not included, called exclusive.
- Brackets, \([\) or \(]\), are used to indicate that an endpoint is included, called inclusive.
See Figure \(\PageIndex{3}\) for a summary of interval notation.
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.
\[\{2,3,4,5,6\} \nonumber\]
Find the domain of the function:
\[\{(−5,4),(0,0),(5,−4),(10,−8),(15,−12)\} \nonumber\]
- Answer
-
\(\{−5, 0, 5, 10, 15\}\)
- Identify the input values.
- Identify any restrictions on the input and exclude those values from the domain.
- Write the domain in interval form, if possible.
Find the domain of the function \(f(x)=x^2−1\).
Solution
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:
\[f(x)=5−x+x^3 \nonumber\]
- Answer
-
\((−\infty,\infty)\)
- Identify the input values.
- Identify any restrictions on the input. If there is a denominator in the function’s formula, set the denominator equal to zero and solve for x . If the function’s formula contains an even root, set the radicand greater than or equal to 0, and then solve.
- Write the domain in interval form, making sure to exclude any restricted values from the domain.
Find the domain of the function \(f(x)=\dfrac{x+1}{2−x}\).
Solution
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.
\[ \begin{align*} 2−x=0 \\[4pt] −x &=−2 \\[4pt] x&=2 \end{align*}\]
Now, we will exclude 2 from the domain. The answers are all real numbers where \(x<2\) or \(x>2\). We can use a symbol known as the union, \(\cup\),to combine the two sets. In interval notation, we write the solution:\((−\infty,2)∪(2,\infty)\).
In interval form, the domain of f is \((−\infty,2)\cup(2,\infty)\).
Find the domain of the function:
\[f(x)=\dfrac{1+4x}{2x−1} \nonumber\]
- Answer
-
\[(−\infty,\dfrac{1}{2})\cup(\dfrac{1}{2},\infty) \nonumber\]
- Identify the input values.
- Since there is an even root, 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.
- The solution(s) are the domain of the function. If possible, write the answer in interval form.
Find the domain of the function:
\[f(x)=\sqrt{7-x} \nonumber .\]
Solution
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.
\[ \begin{align*} 7−x&≥0 \\[4pt] −x&≥−7\\[4pt] x&≤7 \end{align*}\]
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 \(\left(−\infty,7\right]\).
Find the domain of the function
\[f(x)=\sqrt{5+2x}. \nonumber\]
- Answer
-
\[\left[−2.5,\infty\right) \nonumber\]
Yes. For example, the function \(f(x)=-\dfrac{1}{\sqrt{x}}\) has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.
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, \(\{x|10≤x<30\}\) 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\( \{x|10≤x<30\}\) as “the set of x-values such that 10 is less than or equal to x, and x is less than 30.”
Figure \(\PageIndex{4}\) 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
\[\{x| |x|≥3\}=\left(−\infty,−3\right]\cup\left[3,\infty\right)\]
Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form\(\{x|\text{ statement about x}\}\) which is read as, “the set of all x such that the statement about x is true.” For example,
\[\{x|4<x≤12\} \nonumber\]
Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,
\[\left(4,12\right] \nonumber\]
Given a line graph, describe the set of values using interval notation.
- Identify the intervals to be included in the set by determining where the heavy line overlays the real line.
- At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).
- At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).
- Use the union symbol \(\cup\) to combine all intervals into one set.
Describe the intervals of values shown in Figure \(\PageIndex{5}\) using inequality notation, set-builder notation, and interval notation.
Solution
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
\[1≤x≤3 \text{ or }x>5 \nonumber\]
Set-builder Notation
\[\{x|1≤x≤3 \text{ or } x>5\}\nonumber\]
Interval notation
\[[1,3]\cup(5,\infty)\nonumber\]
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 Figure \(\PageIndex{6}\), specify the graphed set in
- words
- set-builder notation
- interval notation
- Answer a
-
Values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3;
- Answer b
-
\(\{x|x≤−2 or −1≤x<3\}\)
- Answer c
-
\(\left(−∞,−2\right]\cup\left[−1,3\right)\)
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 Figure \(\PageIndex{7}\).
We can observe that the graph extends horizontally from −5 to the right without bound, so the domain is \(\left[−5,∞\right)\). The vertical extent of the graph is all range values 5 and below, so the range is \(\left(−∞,5\right]\). 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.
Find the domain and range of the function f whose graph is shown in Figure 1.2.8.
Solution
We can observe that the horizontal extent of the graph is –3 to 1, so the domain of f is \(\left(−3,1\right]\).
The vertical extent of the graph is 0 to –4, so the range is \(\left[−4,0\right)\). See Figure \(\PageIndex{9}\).
Find the domain and range of the function f whose graph is shown in Figure \(\PageIndex{10}\).
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 \(1973≤t≤2008\) and the range as approximately \(180≤b≤2010\).
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 Figure \(\PageIndex{11}\), identify the domain and range using interval notation.
- Answer
-
domain =\([1950,2002]\)
range = \([47,000,000,89,000,000]\)
Can a function’s domain and range be the same?
Yes. For example, the domain and range of the cube root function are both the set of all real numbers.
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.
For the constant function \( f(x)=c\), the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant \(c\), so the range is the set \(\{c\}\) that contains this single element. In interval notation, this is written as \([c,c]\), the interval that both begins and ends with \(c\).
Figure \(\PageIndex{13}\):
Identity function f(x)=x.
For the identity function \(f(x)=x\), there is no restriction on \(x\). Both the domain and range are the set of all real numbers.
For the absolute value function \(f(x)=|x|\), there is no restriction on \(x\). However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.
For the quadratic function \(f(x)=x^2\), the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.
For the cubic function \(f(x)=x^3\), the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.
For the reciprocal function \(f(x)=\dfrac{1}{x}\), we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write\(\{x| x≠0\}\),the set of all real numbers that are not zero.
For the reciprocal squared function \(f(x)=\dfrac{1}{x^2}\),we cannot divide by 0, so we must exclude 0 from the domain. There is also no x that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.
Figure \(\PageIndex{19}\):
Square root function \(f(x)=\sqrt{(x)}\).
For the square root function \(f(x)=\sqrt{x}\), we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number \(x\) is defined to be positive, even though the square of the negative number \(−\sqrt{x}\) also gives us \(x\).
For the cube root function \(f(x)=\sqrt[3]{x}\), the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).
Given the formula for a function, determine the domain and range.
- Exclude from the domain any input values that result in division by zero.
- Exclude from the domain any input values that have nonreal (or undefined) number outputs.
- Use the valid input values to determine the range of the output values.
- Look at the function graph and table values to confirm the actual function behavior.
Find the domain and range of \(f(x)=2x^3−x\).
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 \((−\infty,\infty)\).
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 \((−\infty,0)\cup(0,\infty)\).
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.
\(x+4≥0\) when \(x≥−4\)
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 \(\left[0,\infty\right)\).
Analysis
Figure \(\PageIndex{19}\) represents the function \(f\).
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Find the domain and range of
\(f(x)=\sqrt{−2−x}\).
- Answer
-
domain: \(\left(−\infty,-2\right]\)
range: \(\left[0,\infty\right)\)
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.
- The domain of a function can also be determined by identifying the input values of a function written as an equation.
- Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation.
- For many functions, the domain and range can be determined from a graph.
- An understanding of toolkit functions can be used to find the domain and range of related functions.
- A piecewise function is described by more than one formula.
- A piecewise function can be graphed using each algebraic formula on its assigned subdomain.
Footnotes
1 The Numbers: Where Data and the Movie Business Meet. “Box Office History for Horror Movies.”
http://www.the-numbers.com/market/genre/Horror
. Accessed 3/24/2014
2 www.eia.gov/dnav/pet/hist/Lea...s=MCRFPAK2&f=A.
Glossary
- interval notation
-
a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion
- piecewise function
-
a function in which more than one formula is used to define the output
- set-builder notation
-
a method of describing a set by a rule that all of its members obey; it takes the form {x| statement about x}