Ordered Pairs Domain And Range

ane.2: Domain and Range

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Skills to Develop

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

If you're in the mood for a scary moving picture, 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. Effigy $\PageIndex{1}$ shows the corporeality, in dollars, each of those movies grossed when they were released equally well equally the marketplace 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, nosotros tin can place different independent and dependent variables, and we can analyze the data and the functions to decide the domain and range. For functions defined by an equation rather than by data, determining the domain and range requires a dissimilar kind of assay. In this department, we will investigate methods for determining the domain and range of functions such equally these.

$[Two graphs where the first graph is of the Top-Five Grossing Horror Movies for years 2000-2003 and Market Share of Horror Movies by Year$

Figure $\PageIndex{1}$: Graph of the Tiptop-Five Grossing Horror Movies for years 2000-2003, and a Graph of the Market Share of Horror Movies past Twelvemonth. Based on data compiled by www.the-numbers.com.

In Department 1.1, Functions and Part Annotation, we were introduced to the concepts of domain and range. In this section, we will practice determining domains and ranges for specific functions. Continue in heed that in determining domains and ranges, nosotros demand to consider what is physically possible or meaningful in real-globe examples, such as tickets sales and year in the horror movie example above. Nosotros likewise 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 case, we cannot include whatever input value that leads the states to take an even root of a negative number if the domain and range consist of existent numbers. Or in a function expressed as a caliber, nosotros cannot include any input value in the domain that would pb u.s.a. to split up by 0.

Nosotros can visualize the domain equally a "property area" that contains "raw materials" for a "part auto" and the range equally some other "holding area" for the machine'southward outputs (Figure $\PageIndex{2}$).

$CNX_Precalc_Figure_01_02_002.jpg$

Effigy $\PageIndex{2}$: Diagram of how a part relates two sets

Example $\PageIndex{1}$: Finding the Domain of a Role as a Gear up of Ordered Pairs

Discover the domain and range of the following office: $\{(2, 10),(three, 10),(4, 20),(5, 30),(6, 40)\}$.

Solution

Outset identify the input values. The input value is the first coordinate in an ordered pair. At that place are no restrictions, as the ordered pairs are simply listed. The domain is the prepare of the first coordinates of the ordered pairs.

\[D = \{2,3,four,5,vi\} \nonumber\]

Next identify the output values. The output value is the second coordinate in an ordered pair. The range is the set up of the 2nd coordinates of the ordered pairs. Practise non list the same output twice.

\[R = \{ten,20,30,forty\}\nonumber \]

$try-it.png$ $\PageIndex{1}$

Observe the domain and range of the function:

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

Answer: $D = \{−five, 0, 5, x, 15\}$
$R = \{4, 0, -4, -eight, -12 \} $

Finding the Domain of a Function Defined by an Equation

Permit'due south 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 part. Frequently, finding the domain of such functions involves remembering three different forms. Starting time, 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 role's equation, exclude values from the domain that force the denominator to exist zero. Tertiary, if at that place is an even root, consider excluding values that would make the radicand negative.

We can ofttimes write the domain and range in interval note, 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 betoken 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 0 and less than or equal to 100 and write $\left(0, 100\right]$. Notation: This would not exist quite right, since this interval includes all real numbers between 0 and 100, and one cannot spend $$\sqrt{5}$.

Remember the conventions for sets written using interval annotation:

The smallest term from the interval is written first.
The largest term in the interval is written 2d, following a comma.
All real numbers between the smallest and largest term are included in the set up.
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, chosen inclusive.

Come across Effigy $\PageIndex{three}$ for a summary of interval notation.

Figure $\PageIndex{three}$: Summary of interval notation.

$how-to.png$ Given a function written in equation form, find the domain.

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

Example $\PageIndex{2}$: Finding the Domain of a Office written in equation course

Find the domain of the function $f(10)=x^two−1$.

Solution

The input value, shown by the variable $ten$ in the equation, is squared and and then the event is decreased by ane. Any real number may exist squared then exist decreased past one, so there are no restrictions on the domain of this part. The domain is the set of real numbers.

In interval course the domain of $f$ is $D = (−\infty,\infty)$, or alternatively $D = \mathbb{R}$.

$try-it.png$ $\PageIndex{2}$

Notice the domain of the function:

\[f(x)=five−10+x^3 \nonumber\]

Respond: $D = (−\infty,\infty)$, or $\mathbb{R}$

$how-to.png$ 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 existent 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 $10$ . 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

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

Solution

When there is a denominator, nosotros want to exclude any $x$-values that force the denominator to exist nil. To find these values, we set the denominator equal to 0 and solve for $x$.

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

Now, we will exclude 2 from the domain. The domain consists of all real numbers that do non equal 2, which means $x<two$ or $x>ii$. We can utilise a symbol known as the spousal relationship, $\cup$, to combine the two sets.

$[Line graph of f(x).]$

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

$try-it.png$ $\PageIndex{three}$

Notice the domain of the role:

\[f(x)=\dfrac{one+4x}{2x−i} \nonumber\]

Respond: \[ D = (−\infty,\dfrac{i}{2})\cup(\dfrac{1}{2},\infty) \nonumber\]
Salubrious, to try this problem! Get your professor to requite you extra credit for Try It 1.2.3.

$how-to.png$ Given a function written as an equation that includes an even root, notice the domain

Place where the input values appear in the equation.
Identify any restrictions on the input. Since at that place is an even root, just 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.

Case $\PageIndex{four}$: Finding the Domain of a Function with an Even Root

Observe the domain of the function:

\[f(ten)=\sqrt{vii-ten} \nonumber .\]

Solution

When in that location is an even root in the formula, we exclude whatever real numbers that upshot in a negative number in the radicand.

Set up the radicand greater than or equal to aught and solve for ten.

\[ \brainstorm{align*} vii−10&≥0 \\[5pt] −ten&≥−7\\[5pt] ten&≤7 \cease{align*}\]

Now, nosotros will include any existent number less than or equal to vii in the domain. The domain is $D = \left(−\infty,7\right]$.

$try-it.png$ $\PageIndex{4}$

Find the domain of the function

\[f(10)=\sqrt{x+52}. \nonumber\]

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

$QA.png$ Can in that location be functions in which the domain and range practise not intersect at all?

Yes. For example, the office $f(x)= -\sqrt{\frac{one}{x}}$ has the set up of all positive real numbers as its domain but the set of all negative real numbers every bit its range. As a more than extreme example, a role'south inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers equally outputs, as on an attendance chart), in such cases the domain and range take no elements in common.

Using Different Notations to Specify Domain

In the previous examples, we used interval notation to depict 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-architect notation. For example, $\{10\;|\;x≤x<30\}$ describes the allowed values of $x$ in set up-builder annotation. The braces "$\{\}$" are read as "the prepare of," and the vertical bar "$|$" is read equally "such that," then we would read$ \{x\;|\;10≤ten<30\}$ as "the gear up of $x$-values such that 10 is less than or equal to $x$, and $x$ is less than thirty."

Figure $\PageIndex{4}$ compares inequality notation, ready-builder note, and interval notation.

$clipboard_e29d19ab394e60c22f5cda847985a99e8.png$

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

In prepare annotation, there is a symbol $\loving cup$ to represent "or," and we say we are taking the union of the two sets. Nosotros saw this in before examples, to combine ii unconnected intervals. As another example, we can take the spousal relationship of the sets$\{2,3,5\}$ and $\{iv,five,6\}$: $\{2,3,5\} \cup \{4,6\} = \{2,three,4,5,6\}$. It is the set of all elements that vest 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 2 sets have some elements in common, those elements should be listed just one time in the marriage gear up. For sets of real numbers on intervals, another case of a union is

\[\{ten\;|\; |x|≥3\}=\left(−\infty,−3\right]\loving cup\left[iii,\infty\right)\]

Set-Builder Notation compared with Interval Annotation

Set-builder annotation is a method of specifying a set of elements that satisfy a sure condition. Information technology 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 instance, nosotros can write

\[\{10\;|\;4<10≤12\}, \nonumber\]

which is the set of all $x$ such that $ten$ is greater than iv and less than or equal to 12.

Interval notation is a manner of describing sets that include all real numbers between a lower limit that may or may non be included and an upper limit that may or may not exist included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the prepare, and a parenthesis indicates exclusion from the gear up. For case, the same set only given in prepare-builder annotation can also be written in interval note as

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

$how-to.png$ Given a line graph, describe the set of values using interval notation

Identify the intervals to be included in the prepare past determining where the graphed line overlays the real number line.
At the left end of each interval, utilise "[" for each end value to be included in the set (solid dot) or "(" for each excluded terminate value (open dot).
At the correct cease of each interval, use "]" for each finish value to exist included in the set (filled dot) or ")" for each excluded finish value (open dot).
Use the union symbol $\cup$ to combine all intervals into one set.

Instance $\PageIndex{v}$: Describing Sets on the Real-Number Line

Describe the intervals of values shown in Effigy $\PageIndex{5}$ using inequality note, set-builder notation, and interval notation.

$[Line graph of $one<=x<=3$ and $5$]$

Figure $\PageIndex{5}$: Line graph of $one \leq x \leq iii$ 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 i and less than or equal to three, or a real number greater than v."

Set-builder Notation

\[\{x\;|\;1≤x≤three \text{ or } x>five\}\nonumber\]

Interval notation

\[[1,3]\loving cup(v,\infty)\nonumber\]

Remember that when writing or reading interval annotation, using a square subclass means the boundary value is included in the set. Using a parenthesis means the boundary value is not included in the ready.

$try-it.png$ $\PageIndex{5}$

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

words
set-architect notation
interval annotation

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

Effigy $\PageIndex{6}$: Line graph of $x \leq -2$ or $-1 \leq ten<3$.

Answer: a. Values that are less than or equal to –two, or values that are greater than or equal to –i and less than iii
b. $\{x\;|\;ten≤−2 \mbox{ or } −1≤ten<3\}$
c. $\left(−∞,−2\correct]\cup\left[−1,3\right)$

Finding Domain and Range from Graphs

Another manner to identify the domain of a function is past using graphs. A graph is also an excellent way to identify the range of a function. Considering 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 fix of possible output values, which are shown on the $y$-axis. Proceed in listen that if the graph continues beyond the portion of the graph nosotros tin can encounter, the domain and range may be greater than the visible values. See Figure $\PageIndex{seven}$.

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

Figure $\PageIndex{seven}$: Graph of a polynomial that shows the $ten$-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 $\left[−5,∞\correct)$. The vertical extent of the graph is all values from 5 and beneath, so the range is $\left(−∞,v\right]$. Note that the domain and range are always written from smaller to larger values, or from left to correct for domain, and from the bottom of the graph to the top of the graph for range.

Instance $\PageIndex{6A}$: Finding Domain and Range from a Graph

Find the domain and range of the function $f$ whose graph is shown in Figure 1.2.eight.

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

Figure $\PageIndex{8}$: Graph of a function defined on (-iii, i].

Solution

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

The vertical extent of the graph is 0 to –4. The range is $[−4,0]$. Although 0 is not the output for $ten=-3$; in fact, there is no output for $10=-3$ since -iii is non 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 Effigy $\PageIndex{9}$.

$clipboard_eb98948a6e6f6f592888c40c9c3b92fa4.png$

Effigy $\PageIndex{9}$: Graph of the previous office showing the domain and range

Case $\PageIndex{6B}$: Finding Domain and Range from a Graph of Oil Product

Find the domain and range of the function $f$ whose graph is shown in Figure $\PageIndex{ten}$.

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

Effigy $\PageIndex{10}$: Graph of the Alaska Crude Oil Production where the vertical centrality is thousand barrels per day and the horizontal axis is years (credit: modification of piece of work by the U.S. Energy Data Assistants)

Solution:

The input quantity along the horizontal axis is "years," which nosotros represent with the variable $t$ for time. The output quantity is "thousands of barrels of oil per mean solar day," which nosotros represent with the variable $b$ for barrels. The graph may keep 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 equally approximately $180≤b≤2010$. Note: This is not really correct, considering 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 exercise not fall exactly on the grid lines.

$try-it.png$ $\PageIndex{vi}$

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

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

Figure $\PageIndex{11}$: Graph of Earth Population Increase where the y-centrality represents millions of people and the x-centrality represents the yr.

Answer

domain =$[1950,2000]$

range = $[4.seven\times 10^seven,9.0\times x^vii]$

$QA.png$ Can a function's domain and range be the aforementioned?

Yes. For example, the domain and range of the cube root function are both the fix of all real numbers.

Finding Domains and Ranges of the Toolkit Functions

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

$clipboard_ee654e5a0b435537278e5d4e3dfcb7224.png$

Effigy $\PageIndex{12}$: Constant office $f(10)=c$.

For the constant part$ f(10)=c$, the domain consists of all real numbers; there are no restrictions on the input. The only output value is the abiding $c$, so the range is the ready $\{c\}$ that contains this single element. In interval notation, this could be written as $[c,c]$, the interval that both begins and ends with $c$.

$[Identity function f(x)=x.]$
Figure $\PageIndex{xiii}$: Identity role $f(x)=ten$.

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.

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

Figure $\PageIndex{14}$: Absolute role $f(x)=|10|$.

For the absolute value function $f(10)=|x|$, there is no restriction on $x$. Notwithstanding, because accented value is defined as a distance from 0, the output tin can but be greater than or equal to 0.

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

Figure $\PageIndex{15}$: Quadratic function $f(x)=x^2$.

For the quadratic office $f(ten)=x^two$, the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does non include any negative values for the range, the range is only nonnegative real numbers.

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

Effigy $\PageIndex{16}$: Cubic function $f(x)=x^3$.

For the cubic function $f(x)=x^three$, the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The aforementioned applies to the vertical extent of the graph, and so the domain and range include all real numbers.

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

Figure $\PageIndex{17}$: Reciprocal function $f(x)=\dfrac{one}{ten}$.

For the reciprocal office $f(x)=\dfrac{i}{10}$, we cannot divide by 0, and so we must exclude 0 from the domain. Further, 1 divided past whatever value tin never be 0, and so the range also will not include 0. In set-architect notation, we could also write $\{x| x≠0\}$, the set up of all real numbers that are not zero, for both the domain and the range.

$[Reciprocal squared function ...]$

Figure $\PageIndex{18}$: Reciprocal squared office $f(x)=\dfrac{1}{ten^2}$

For the reciprocal squared role $f(x)=\dfrac{1}{x^two}$,we cannot divide by 0, and then we must exclude 0 from the domain. In that location is as well no x that tin can give an output of 0, and so 0 is excluded from the range as well. Note that the output of this function is ever positive due to the square in the denominator, so the range includes just positive numbers.

$[Square root function f(x)=sqrt(x).]$
Effigy $\PageIndex{xix}$: Square root function $f(x)=\sqrt{x}$.

For the foursquare root function $f(x)=\sqrt{x}$, we cannot accept 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 $10$ as well has a negative foursquare root, denoted $-\sqrt{x}$.

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

Figure $\PageIndex{twenty}$: Cube root part $f(x)=\sqrt[3]{10}$.

For the cube root role $f(x)=\sqrt[three]{x}$, the domain and range include all real numbers. Notation that in that location is no trouble taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).

$how-to.png$ Given the formula for a function, determine the domain and range

Exclude from the domain any input values that result in division past zero.
Exclude from the domain any input values that have nonreal (or undefined) number outputs.
If possible, use the valid input values to make up one's mind the range of the output values.
Look at the function graph and/or tabular array values to confirm the actual function behavior.

Case $\PageIndex{vii}$: Finding the Domain Using Toolkit Functions

Discover the domain of $f(ten)=2x^iii−10$.

Solution

This function uses ii Toolkit Functions equally building blocks. At that place are no restrictions on the domain, every bit any real number may exist cubed and so subtracted from the result.

The domain is $(−\infty,\infty)$.

Example $\PageIndex{8}$: Finding the Domain

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

Solution

Nosotros cannot evaluate the part at $x=−one$ considering division by zero is undefined. The domain is $(−\infty,−i)\cup(−i,\infty)$.

Example $\PageIndex{ix}$: Finding the Domain

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

Solution

Nosotros cannot have the foursquare root of a negative number, so the value inside the radical must be nonnegative.

$10+4≥0$ when $x≥−4$

The domain of $f(ten)$ is $[−4,\infty)$.

We can also discover the range, past comparison this function with the Toolkit Function $one thousand(10)=\sqrt{ten}$. Nosotros know that $f(−4)=0$, and the function value increases as $ten$ increases without whatever upper limit. We conclude that the range of $f$ is $\left[0,\infty\right)$.

Assay

Figure $\PageIndex{xix}$ represents the graph of the part $f$.

$<div data-mt-source="1"><img class="internal" alt="" data-cke-saved-src="http://mathwiki.ucdavis.edu/@api/deki/files/892/CNX_Precalc_Figure_01_02_020.jpg" src="http://mathwiki.ucdavis.edu/@api/deki/files/892/CNX_Precalc_Figure_01_02_020.jpg" height="237" width="350"></div>$

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

$try-it.png$ $\PageIndex{7}$

Find the domain of

$f(x)=\sqrt{−2−10}$.

Respond: domain: $\left(−\infty,two\correct]$

Graphing Piecewise-Defined Functions

Sometimes, we come beyond a function that requires more one formula in social club to obtain the given output. For example, in the toolkit functions, we introduced the accented value function $f(x)=|x|$. With a domain of all existent numbers and a range of values greater than or equal to 0, absolute value can be divers 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 crave the output to be greater than or equal to 0.

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

\[f(10)=x \mbox{ if } ten≥0 \nonumber \]

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

\[f(10)=−x \mbox{ if } 10<0 \nonumber\]

Considering this requires 2 unlike processes or pieces, the absolute value role is an instance of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over unlike pieces of the domain.

We use piecewise functions to depict situations in which a dominion or relationship changes every bit the input value crosses certain "boundaries." Tax brackets are a real-world case of piecewise functions. For case, consider a uncomplicated tax organisation in which incomes upwardly to $10,000 are taxed at 10%, and any additional income is taxed at twenty%. The tax $T$ on a total income S would exist

\[T(S)=\begin{cases}0.1S \mbox{ if } S≤$10,000 \\ \$k+0.ii(South−$10,000) \mbox{ if } South>$10,000.\end{cases} \nonumber\]

Can you give a clear explanation of the reason for the formula of each piece of this income revenue enhancement function? If so, write information technology up and give information technology to your teacher for Extra Credit .

Piecewise Function

A piecewise part is a unmarried function in which more than 1 formula is used to define the output. Each formula has its own domain; however, the domain of the role is the spousal relationship of all these smaller domains. We notate this idea as follows:

\[f(ten)= \begin{cases} \text{formula 1} & \text{if 10 is in domain ane} \\ \text{formula 2} &\text{if x is in domain 2} \\ \text{formula 3} &\text{if 10 is in domain 3}\end{cases} \nonumber\]

In piecewise note, the absolute value role is

\[|x|= \begin{cases} x & \text{if $x \geq 0$} \\ -10 &\text{if $x<0$} \finish{cases} \nonumber\]

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

$how-to.png$ Given a piecewise part, write the formula and identify the domain for each interval

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

Example $\PageIndex{ten}$: Writing a Piecewise Function

A museum charges $5 per person for a guided tour with a grouping of 1 to 9 people or a stock-still $50 fee for a group of ten or more than people. Write a office relating the number of people, $n$, to the cost, $C$.

Solution

Two unlike formulas will be needed. For $northward$-values under ten, $C=5n$. For values of n that are 10 or greater, $C=50$.

\[C(north)= \begin{cases} 5n & \text{if $n < 10$} \\ fifty &\text{if $due north\geq 10$} \stop{cases} \nonumber\]

Assay

The function is represented equally a graph in Figure $\PageIndex{20}$. The graph is a straight line with gradient of v from $northward=0$ to $due north=ten$ and a constant later that. In this example, the two formulas agree at the meeting bespeak where $n=x$, only not all piecewise functions take this belongings.

$[Graph of C(n).]$

Figure $\PageIndex{20}$

Note: Looking at the piecewise definition of role $C$, it seems that the domain is $D = (-\infty, \infty)$. Looking at the graph, information technology 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 lot think the domain is? If yous think you know, write an explanation and give it to your instructor for Actress Credit .

Instance $\PageIndex{xi}$: Working with a Piecewise Part

A cell phone visitor uses the function below to determine the price $C$ in dollars for $yard$ gigabytes of information transfer.

\[C(thousand)= \begin{cases} 25 & \text{if $0<1000<two$} \\ 25+10(thousand-2) &\text{if $g\geq2$} \end{cases} \nonumber\]

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

Soltuion

To find the toll of using 1.five gigabytes of information, $C(1.five)$, we first look to run across which part of the domain our input falls in. Because 1.5 is less than two, we use the start formula.

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

To find the cost of using 4 gigabytes of data, C(four), we see that our input of 4 is greater than 2, so we use the second formula.

\[C(four)=25+10(4−2)=$45 \nonumber\]

Assay

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 $g=two$. We plot the graphs for the different formulas on a common gear up of axes, making sure each formula is applied on its proper domain.

$[Graph of C(g)]$

Figure $\PageIndex{21}$

Annotation: Calculate the slope of the straight line defined as the 2nd slice of the function $C$, and give your work to your instructor for Extra Credit .

$how-to.png$ Given a piecewise function, sketch a graph

Bespeak on the x-centrality the boundaries defined by the intervals on each piece of the domain.
For each piece of the domain, graph on that interval using the respective equation pertaining to that piece. Practise not graph two functions over i interval because information technology would violate the criteria of a function.

Example $\PageIndex{12}$: Graphing a Piecewise Function

Sketch a graph of the part.

\[f(x)= \brainstorm{cases} x^ii & \text{if $x \leq 1$} \\ 3 &\text{if $i<x\leq2$} \\ 10 &\text{if $x>2$} \finish{cases} \nonumber\]

Solution

\[f(x)= \begin{cases} x^2 & \text{if $10 \leq ane$} \\ iii &\text{if $one<x\leq2$} \\ x &\text{if $x>ii$} \end{cases} \nonumber\]

Each of the component functions is from our library of toolkit functions, then nosotros know their shapes. Nosotros can imagine graphing each office and and so limiting the graph to the indicated domain. At the endpoints of the domain, nosotros draw open circles to indicate where the endpoint is non included because of a less-than or greater-than inequality; we depict a closed circumvolve where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality.

Effigy $\PageIndex{twenty}$ 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 $\PageIndex{20}$: Graph of each part of the piece-wise function f(x)

(a)$ f(x)=ten^2$ if $x≤1$; (b) $f(10)=three$ if $i< x≤2$; (c) $f(10)=x$ if $x>two$

Now that we have sketched each slice individually, we combine them in the aforementioned coordinate plane. Come across Figure $\PageIndex{21}$.

$[Graph of the entire function.]$

Effigy $\PageIndex{21}$: Graph of the entire function.

Analysis

Note that the graph does laissez passer the vertical line test even at $x=ane$ and $10=2$ because the points $(ane,iii)$ and $(two,2)$ are not part of the graph of the function, though $(i,1)$ and $(2, 3)$ are.

$try-it.png$ $\PageIndex{8}$

Graph the following piecewise function.

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

Answer

$[Graph of f(x).]$

Effigy $\PageIndex{22}$

$QA.png$ Can more than than one formula from a piecewise office exist 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 non cause united states to effort an undefined mathematical functioning, such as dividing by zero or taking the square root of a negative number.
The domain of a office can exist adamant by listing the input values of a set of ordered pairs.
The domain of a function can too exist determined past identifying the input values of a function written as an equation.
Interval values represented on a number line tin 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 office is described by more than ane formula.
A piecewise function can be graphed using each algebraic formula on its assigned subdomain.

Glossary

interval note: a method of describing a ready that includes all numbers betwixt 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 role in which more than one formula is used to define the output

set up-architect annotation: a method of describing a fix past a dominion that all of its members obey; information technology takes the form ${x| statement near x}$