- Last updated

- Save as PDF

- Page ID
- 15566

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

\( \newcommand{\vectorC}[1]{\textbf{#1}}\)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}}\)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}\)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

Skills to Develop

- Find the domain of a function defined by an equation.
- Find the domain and range of a function from a graph.
- Graph piecewise-defined functions.

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 \(\PageIndex{1}\) shows the amount, in dollars, each of those movies grossed when they were released as well as the market share, as a percent, for horror movies in general by year. We could use these data to create a function of the amount each movie earned or the market percentage 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. For functions defined by an equation rather than by data, determining the domain and range requires a different kind of analysis. In this section, we will investigate methods for determining the domain and range of functions such as these.

**Figure \(\PageIndex{1}\): **Graph of the Top-Five Grossing Horror Movies for years 2000-2003, and a Graph of the Market Share of Horror Movies by Year**. ***Based on data compiled by www.the-numb*ers.com.

In Section 1.1, 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. In this course, unless specifically stated otherwise, we are interested in functions whose inputs and outputs are always real numbers. 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 quotient, 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 outputs (Figure \(\PageIndex{2}\)).

**Figure \(\PageIndex{2}\): **Diagram of how a function relates two sets

Example \(\PageIndex{1}\): Finding the Domain of a Function as a Set of Ordered Pairs

Find the domain and range 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.

\[D = \{2,3,4,5,6\} \nonumber\]

Next identify the output values. The output value is the second coordinate in an ordered pair. The range is the set of the second coordinates of the ordered pairs. Do not list the same output twice.

\[R = \{10,20,30,40\}\nonumber \]

\(\PageIndex{1}\)

Find the domain and range of the function:

\[\{(−5,4),(0,0),(5,−4),(10,−8),(15,−12)\} \nonumber\]

**Answer**-
\(D = \{−5, 0, 5, 10, 15\}\)

\(R = \{4, 0, -4, -8, -12 \} \)

## Finding the Domain of a Function Defined by an Equation

Let’s turn our attention to finding the domain of a function whose equation is provided. Finding the range is harder. We will see later on in this section how to find the range if we have a graph of the function. 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 from 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.

We can often 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 might express the interval that is more than 0 and less than or equal to 100 and write \(\left(0, 100\right]\).** Note: **This would not be quite correct, since this interval includes **all** real numbers between 0 and 100, and one cannot spend $\(\sqrt{5}\).

Recall the conventions for sets written using interval notation:

- The smallest term from the interval is written first.
- The largest term in the interval is written second, following a comma.
- All real numbers between the smallest and largest term are included in the set.
- 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.

**Figure \(\PageIndex{3}\): **Summary of interval notation.

Given a function written in equation form, find the domain.

- Identify where the input values appear in the equation.
- Begin with the set of all real numbers as the possible domain.
- Identify any restrictions on the input and exclude those values from the set of real numbers.
- Write the domain in interval form, if possible.

Example \(\PageIndex{2}\): Finding the Domain of a Function written in equation form

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 decreased by one. Any real number may be squared and then be decreased 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 \(D = (−\infty,\infty)\), or alternatively \(D = \mathbb{R}\).

\(\PageIndex{2}\)

Find the domain of the function:

\[f(x)=5−x+x^3 \nonumber\]

**Answer**-
\(D = (−\infty,\infty)\), or \(\mathbb{R}\)

Given a function written in an equation form that includes a fraction, find the domain

- Identify where the input values appear in the equation.
- Begin with the set of all real numbers as the possible domain.
- Identify any restrictions on the input. If the denominator contains an input value, set the denominator equal to zero and solve for \(x\) . The domain must be restricted to
**exclude**all solutions to this equation. - Write the domain in interval form, making sure to exclude any restricted values from the domain.

Example \(\PageIndex{3}\): Finding the Domain of a Function Involving a Denominator

Find the domain of the function \(f(x)=\dfrac{x+1}{2−x}\).

**Solution**

When there is a denominator, we want to exclude any \(x\)-values that force the denominator to be zero. To find these values, we set the denominator equal to 0 and solve for \(x\).

\[ \begin{align*} 2−x=0 \\[5pt] −x &=−2 \\[5pt] x&=2 \end{align*}\]

Now, we will exclude 2 from the domain. The domain consists of all real numbers that do not equal 2, which means \(x<2\) or \(x>2\). We can use a symbol known as the union, \(\cup\), to combine the two sets.

In interval form, the domain of \(f\) is \(D = (−\infty,2)\cup(2,\infty)\).

\(\PageIndex{3}\)

Find the domain of the function:

\[f(x)=\dfrac{1+4x}{2x−1} \nonumber\]

**Answer**-
\[ D = (−\infty,\dfrac{1}{2})\cup(\dfrac{1}{2},\infty) \nonumber\]

Good for you, to try this problem! Get your professor to give you extra credit for**Try It 1.2.3**.

Given a function written as an equation that includes an even root, find the domain

- Identify where the input values appear in the equation.
- Identify any restrictions on the input. Since there is an even root, only include inputs that do not 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.

Example \(\PageIndex{4}\): Finding the Domain of a Function with an Even Root

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.

\[ \begin{align*} 7−x&≥0 \\[5pt] −x&≥−7\\[5pt] x&≤7 \end{align*}\]

Now, we will include any real number less than or equal to 7 in the domain. The domain is \(D = \left(−\infty,7\right]\).

\(\PageIndex{4}\)

Find the domain of the function

\[f(x)=\sqrt{x+52}. \nonumber\]

**Answer**-
\[\left[−52,\infty\right) \nonumber\]

** **Can there be functions in which the domain and range do not intersect at all?

*Yes. For example, the function \(f(x)= -\sqrt{\frac{1}{x}}\) 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 Different Notations to Specify Domain

In the previous examples, we used interval notation 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 domain in **set-builder notation**. For example, \(\{x\;|\;10≤x<30\}\) describes the allowed values 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.

**Figure \(\PageIndex{4}\): **Summary of notations for inequalities, set-builder, and interval.

In set notation, there is a symbol \(\cup\) to represent “or,” and we say we are taking the union of the two sets. We saw this in earlier examples, to combine two unconnected intervals. As another example, we can take the union of the sets\(\{2,3,5\}\) and \(\{4,5,6\}\): \(\{2,3,5\} \cup \{4,6\} = \{2,3,4,5,6\}\). 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 compared with Interval Notation

*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, we can write

\[\{x\;|\;4<x≤12\}, \nonumber\]

which is the set of all \(x\) such that \(x\) is greater than 4 and less than or equal to 12.

*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, the same set just given in set-builder notation can also be written in interval notation as

\[\left(4,12\right]. \nonumber\]

** **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 graphed line overlays the real number line.
- At the left end of each interval, use "[" for 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 "]" for 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.

Example \(\PageIndex{5}\): Describing Sets on the Real-Number Line

Describe the intervals of values shown in Figure \(\PageIndex{5}\) using inequality notation, set-builder notation, and interval notation.

**Figure \(\PageIndex{5}\): **Line graph of \(1 \leq x \leq 3\) or \(5<x\).

**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.”

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 value is included in the set. Using a parenthesis means the boundary value is not included in the set.

\(\PageIndex{5}\)

Given Figure \(\PageIndex{6}\), specify the graphed set in

- words
- set-builder notation
- interval notation

**Figure \(\PageIndex{6}\): **Line graph of \(x \leq -2\) or \(-1 \leq x<3\).

**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

b. \(\{x\;|\;x≤−2 \mbox{ or } −1≤x<3\}\)

c. \(\left(−∞,−2\right]\cup\left[−1,3\right)\)

## Finding Domain and Range from Graphs

Another way to identify the domain of a function is by using graphs. A graph is also an excellent way to identify the range of a function. 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 \(\PageIndex{7}\).

**Figure \(\PageIndex{7}\): **Graph of a polynomial that shows the \(x\)-axis contains the domain and the \(y\)-axis contains the range

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 values from 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 \(\PageIndex{6A}\): 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.

**Figure \(\PageIndex{8}\):***Graph of a function defined on (-3, 1].*

**Solution**

Observe that the horizontal extent of the graph is –3 to 1, not including \(x=-3\) but including \(x=1\), so the domain of f is \(\left(−3,1\right]\).

The vertical extent of the graph is 0 to –4. The range is \([−4,0]\). Although 0 is not the output for \(x=-3\); in fact, there is no output for \(x=-3\) since -3 is not part of the domain; we can see that the point \((0,0)\) is on the graph, which means \(f(0)=0\), and so 0 **is** in the range. See Figure \(\PageIndex{9}\).

**Figure \(\PageIndex{9}\): **Graph of the previous function showing the domain and range

Example \(\PageIndex{6B}\): 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 \(\PageIndex{10}\).

*Figure \(\PageIndex{10}\): **Graph of the Alaska Crude Oil Production where the vertical axis is thousand barrels per day and the horizontal axis is years (credit: modification of work by the U.S. Energy Information Administration)*

**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 \(180≤b≤2010\). **Note: **This is not really correct, because not every real number is included in the domain, nor in the range. The graph represents a *mathematical model* of a real-life situation.

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.

\(\PageIndex{6}\)

Given Figure \(\PageIndex{11}\), identify the approximate domain and range using interval notation.

**Figure \(\PageIndex{11}\): **Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year.

**Answer**-
domain =\([1950,2000]\)

range = \([4.7\times 10^7,9.0\times 10^7]\)

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.

**Figure \(\PageIndex{12}\): **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 \(\{c\}\) that contains this single element. In interval notation, this could be 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.

*Figure \(\PageIndex{14}\): **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.

*Figure \(\PageIndex{15}\): **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.

*Figure \(\PageIndex{16}\): **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.

*Figure \(\PageIndex{17}\): **Reciprocal function \(f(x)=\dfrac{1}{x}\).*

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 both the domain and the range.

*Figure \(\PageIndex{18}\): **Reciprocal squared function \(f(x)=\dfrac{1}{x^2}\)*

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 symbol \(\sqrt{x}\) is defined to be the positive square root of \(x\), even though \(x\) also has a negative square root, denoted \(-\sqrt{x}\).

*Figure \(\PageIndex{20}\): **Cube root function *\(f(x)=\sqrt[3]{x}\)*.*

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.
- If possible, use the valid input values to determine the range of the output values.
- Look at the function graph and/or table values to confirm the actual function behavior.

Example \(\PageIndex{7}\)**: **Finding the Domain Using Toolkit Functions

Find the domain of \(f(x)=2x^3−x\).

**Solution**

This function uses two Toolkit Functions as building blocks. There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

The domain is \((−\infty,\infty)\).

Example \(\PageIndex{8}\): Finding the Domain

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

**Solution**

We cannot evaluate the function at \(x=−1\) because division by zero is undefined. The domain is \((−\infty,−1)\cup(−1,\infty)\).

Example \(\PageIndex{9}\): Finding the Domain

Find the domain 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 can also find the range, by comparing this function with the Toolkit Function \(g(x)=\sqrt{x}\). 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 graph of the function \(f\).

**Figure \(\PageIndex{19}\): **Graph of a square root function which has been shifted to \((-4, 0)\).

\(\PageIndex{7}\)

Find the domain of

\(f(x)=\sqrt{−2−x}\).

**Answer**-
domain: \(\left(−\infty,2\right]\)

## 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.

\[f(x)=x \mbox{ if } x≥0 \nonumber \]

If we input a negative value, the output is the opposite of the input. For example, \(f(-3) = -(-3) = 3\).

\[f(x)=−x \mbox{ if } x<0 \nonumber\]

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.” Tax brackets are a 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 \(T\) on a total income S would be

\[T(S)=\begin{cases}0.1S \mbox{ if } S≤$10,000 \\ \$1000+0.2(S−$10,000) \mbox{ if } S>$10,000.\end{cases} \nonumber\]

Can you give a clear explanation of the reason for the formula of each piece of this income tax function? If so, write it up and give it to your instructor for * Extra Credit*.

Piecewise Function

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

\[f(x)= \begin{cases} \text{formula 1} & \text{if x is in domain 1} \\ \text{formula 2} &\text{if x is in domain 2} \\ \text{formula 3} &\text{if x is in domain 3}\end{cases} \nonumber\]

In piecewise notation, the absolute value function is

\[|x|= \begin{cases} x & \text{if $x \geq 0$} \\ -x &\text{if $x<0$} \end{cases} \nonumber\]

The domain for the absolute value function is the union of \([0, \infty)\) and \((-\infty, 0)\); or, all real numbers.

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

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

Example \(\PageIndex{10}\): 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, \(C=50\).

\[C(n)= \begin{cases} 5n & \text{if $n < 10$} \\ 50 &\text{if $n\geq 10$} \end{cases} \nonumber\]

**Analysis**

The function is represented as a graph in Figure \(\PageIndex{20}\). The graph is a straight line with slope of 5 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.

**Figure \(\PageIndex{20}\)**

**Note:** Looking at the piecewise definition of function \(C\), it seems that the domain is \(D = (-\infty, \infty)\). Looking at the graph, it seems that the domain for \(C\) is \(D=[0, \infty)\). Given that this function is a mathematical model for a real-life situation, what do you think the domain is? If you think you know, write an explanation and give it to your instructor for ** Extra Credit**.

Example \(\PageIndex{11}\): 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.

\[C(g)= \begin{cases} 25 & \text{if $0<g<2$} \\ 25+10(g-2) &\text{if $g\geq2$} \end{cases} \nonumber\]

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.

\[C(1.5)=$25 \nonumber\]

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.

\[C(4)=25+10(4−2)=$45 \nonumber\]

**Analysis**

The function is represented in Figure \(\PageIndex{21}\). We can see where the function changes from a constant to a linear function with positive slope 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.

**Figure \(\PageIndex{21}\)**

**Note**: Calculate the slope of the straight line defined as the second piece of the function \(C\), and give your work to your instructor for ** Extra Credit**.

Given a piecewise function, sketch a graph

- Indicate on the x-axis the boundaries defined by the intervals on each piece of the domain.
- 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 \(\PageIndex{12}\): Graphing a Piecewise Function

Sketch a graph of the function.

\[f(x)= \begin{cases} x^2 & \text{if $x \leq 1$} \\ 3 &\text{if $1<x\leq2$} \\ x &\text{if $x>2$} \end{cases} \nonumber\]

**Solution**

\[f(x)= \begin{cases} x^2 & \text{if $x \leq 1$} \\ 3 &\text{if $1<x\leq2$} \\ x &\text{if $x>2$} \end{cases} \nonumber\]

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 \(\PageIndex{20}\) shows the three components of the piecewise function graphed on separate coordinate systems.

**Figure \(\PageIndex{20}\):** Graph of each part of the piece-wise function f(x)

(a)\( f(x)=x^2\) if \(x≤1\); (b) \(f(x)=3\) if \(1< x≤2\); (c) \(f(x)=x\) if \(x>2\)

Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure \(\PageIndex{21}\).

**Figure \(\PageIndex{21}\): **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.

\(\PageIndex{8}\)

Graph the following piecewise function.

\[f(x)= \begin{cases} x^3 & \text{if $x < -1$} \\ -2 &\text{if $-1<x<4$} \\ \sqrt{x} &\text{if $x>4$} \end{cases} \nonumber\]

**Answer**-
**Figure \(\PageIndex{22}\)**

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

- 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 http://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 \({x| statement about x}\)

## Contributors

Jay Abramson (Arizona State University) with contributing authors. Textbook content produced byOpenStax Collegeis licensed under aCreative Commons Attribution License 4.0license.Download for free athttps://openstax.org/details/books/precalculus.