3.3: Domain and Range

If you’re in the mood for a scary movie, you may want to check out one of the five most popular horror movies of all time—I am Legend, Hannibal, The Ring, The Grudge, and The Conjuring. Figure \(\PageIndex\) shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions such as these.

[Two graphs where the first graph <a href=is of the Top-Five Grossing Horror Movies for years 2000-2003 and Market Share of Horror Movies by Year" />

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 \(\PageIndex\)).

alt

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:

See Figure \(\PageIndex\) for a summary of interval notation.

[Line graph of f(x).]

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\(\\) and \(\\) is the set \(\\). 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

Set-Builder Notation and Interval Notation

Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form\(\\>\) which is read as, “the set of all x such that the statement about x is true.” For example,

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,

Given a line graph, describe the set of values using interval notation.

  1. Identify the intervals to be included in the set by determining where the heavy line overlays the real line.
  2. 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).
  3. 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).
  4. Use the union symbol \(\cup\) to combine all intervals into one set.

Example \(\PageIndex\): Describing Sets on the Real-Number Line

Describe the intervals of values shown in Figure \(\PageIndex\) using inequality notation, set-builder notation, and interval notation.

[Line graph of \(1<=x<=3\) and \(5\)]

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.”

\[1≤x≤3 \text< or >x>5 \nonumber\]

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 \(\PageIndex\), specify the graphed set in

[Line graph of -2<=x, -1<=x<3.]

  1. words
  2. set-builder notation
  3. 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

Answer c

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 \(\PageIndex\).

[Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range]

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.

Example \(\PageIndex\): Finding Domain and Range from a Graph

Find the domain and range of the function f whose graph is shown in Figure 1.2.8.

[Graph of a function from (-3, 1].]

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 \(\PageIndex\).

[Graph of the previous function shows the domain and range.]

Example \(\PageIndex\): Finding Domain and Range from a Graph of Oil Production

Find the domain and range of the function f whose graph is shown in Figure \(\PageIndex\).

[Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per day and the -axis is the years.]

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 \(\PageIndex\), identify the domain and range using interval notation.

[Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year.]

Answer

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.

[Constant function f(x)=c.]

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 \(\\) that contains this single element. In interval notation, this is written as \([c,c]\), the interval that both begins and ends with \(c\).

[Identity function f(x)=x.]


Figure \(\PageIndex\): 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.

[Absolute function f(x)=|x|.]

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.

[quadratic function f(x)=x^2]

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.

[Cubic function f(x)-x^3.]

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.

[Reciprocal function f(x)=1/x.]

For the reciprocal function \(f(x)=\dfrac\), 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\(\\),the set of all real numbers that are not zero.

[Reciprocal squared function . ]

For the reciprocal squared function \(f(x)=\dfrac\),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.

[Square root function f(x)=sqrt(x).]


Figure \(\PageIndex\): Square root function \(f(x)=\sqrt\).

For the square root function \(f(x)=\sqrt\), 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 \(−\sqrt\) also gives us \(x\).

[Cube root function f(x)=x^(1/3).]

For the cube root function \(f(x)=\sqrt[3]\), 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.

  1. Exclude from the domain any input values that result in division by zero.
  2. Exclude from the domain any input values that have nonreal (or undefined) number outputs.
  3. Use the valid input values to determine the range of the output values.
  4. Look at the function graph and table values to confirm the actual function behavior.

Finding the Domain and Range Using Toolkit Functions

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 \((−\infty,\infty)\).

Example \(\PageIndex\): Finding the Domain and Range

Find the domain and range of \(f(x)=\frac\).

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 \(\PageIndex\): Finding the Domain and Range

Find the domain and range of \(f(x)=2 \sqrt\).

Solution

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

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 \(\PageIndex\) represents the function \(f\).

<div data-mt-source=

" width="" height="" />

Find the domain and range of

Answer

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.

\[ f(x)=x \; \text < if >\; x≥0 \nonumber \]

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

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 \(S≤$10,000\) and \($1000+0.2(S−$10,000)\) if \(S>$10,000\).

A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:

In piecewise notation, the absolute value function is

Given a piecewise function, write the formula and identify the domain for each interval.

  1. Identify the intervals for which different rules apply.
  2. Determine formulas that describe how to calculate an output from an input in each interval.
  3. Use braces and if-statements to write the function.

Example \(\PageIndex\): Writing a Piecewise Function

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, \(C\).

Solution

Two different formulas will be needed. For \(n\)-values under 10, \(C=5n\). For values of n that are 10 or greater, \(C=50\).

Analysis

The function is represented in Figure \(\PageIndex\). The graph is a diagonal line from \(n=0\) to \(n=10\) and a constant after that. In this example, the two formulas agree at the meeting point where \(n=10\), but not all piecewise functions have this property.

[Graph of C(n).]

Example \(\PageIndex\): Working with a Piecewise Function

A cell phone company uses the function below to determine the cost, C, in dollars for g gigabytes of data transfer.

Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

Soltuion

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.

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.

Analysis

The function is represented in Figure \(\PageIndex\). We can see where the function changes from a constant to a shifted and stretched identity at \(g=2\). We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.

[Graph of C(g)]

Given a piecewise function, sketch a graph.

  1. Indicate on the x-axis the boundaries defined by the intervals on each piece of the domain.
  2. For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of a function.

Example \(\PageIndex\): Graphing a Piecewise Function

Sketch a graph of the function.

Solution

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.

Figure \(\PageIndex\) shows the three components of the piecewise function graphed on separate coordinate systems.

[Graph of each part of the piece-wise function f(x)]


Figure \(\PageIndex\): Graph of each part of the piece-wise function f(x)

Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure \(\PageIndex\).

[Graph of the entire function.]

Analysis

Note that the graph does pass the vertical line test even at \(x=1\) and \(x=2\) because the points \((1,3)\) and \((2,2)\) are not part of the graph of the function, though \((1,1)\) and \((2, 3)\) are.

Graph the following piecewise function.

[Graph of f(x).]

Answer

Can more than one formula from a piecewise function be applied to a value in the domain?

No. Each value corresponds to one equation in a piecewise formula.

Key Concepts

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

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