Domain And Range Piecewise Functions

2.3: Domain and Range, Piecewise Functions

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Learning Objectives

• Find the domain of a function defined by an equation.
• Graph piecewise-defined functions.

If yous're in the mood for a scary movie, y'all may want to cheque out 1 of the five most popular horror movies of all time—I am Legend, Hannibal, The Band, The Grudge, and The Conjuring. Figure \(\PageIndex{one}\) shows the corporeality, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general past twelvemonth. Discover that we tin can employ the data to create a part of the amount each motion-picture show earned or the full ticket sales for all horror movies by year. In creating various functions using the data, we can identify unlike contained and dependent variables, and we can analyze the data and the functions to decide the domain and range. In this section, we will investigate methods for determining the domain and range of functions such as these.

Effigy \(\PageIndex{1}\): 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-numbers.com.

Finding the Domain of a Part Defined by an Equation

In Functions and Role Notation, we were introduced to the concepts of domain and range. In this department, 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 moving picture case above. Nosotros also need to consider what is mathematically permitted. For instance, we cannot include any input value that leads us to take an fifty-fifty root of a negative number if the domain and range consist of real numbers. Or in a function expressed every bit a formula, we cannot include whatever input value in the domain that would pb us to split up by 0.

We can visualize the domain every bit a "property area" that contains "raw materials" for a "function car" and the range as another "property area" for the machine'due south products (Figure \(\PageIndex{ii}\)).

Figure \(\PageIndex{two}\): Diagram of how a function relates two relations

We tin write the domain and range in interval notation, which uses values within brackets to depict a set of numbers. In interval note, we use a foursquare bracket [ when the ready includes the endpoint and a parenthesis ( to bespeak 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 demand to express the interval that is more than 0 and less than or equal to 100 and write \(\left(0, 100\right]\). Nosotros will discuss interval note in greater detail later.

Let's turn our attending to finding the domain of a part whose equation is provided. Often, finding the domain of such functions involves remembering iii different forms. First, if the office has no denominator or an even root, consider whether the domain could be all existent numbers. 2nd, if there is a denominator in the function'due south equation, exclude values in the domain that force the denominator to be cipher. Third, if there is an even root, consider excluding values that would make the radicand negative.

Earlier nosotros brainstorm, 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, post-obit a comma.
• Parentheses, \((\) or \()\), are used to signify that an endpoint is not included, called sectional.
• Brackets, \([\) or \(]\), are used to indicate that an endpoint is included, chosen inclusive.

See Effigy \(\PageIndex{3}\) for a summary of interval notation.

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

Example \(\PageIndex{i}\): Finding the Domain of a Function every bit a Set of Ordered Pairs

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

Solution

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

\[\{ii,three,4,5,half dozen\} \nonumber\]

Exercse \(\PageIndex{ane}\)

Detect the domain of the role:

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

Answer

\(\{−5, 0, 5, 10, 15\}\)

How To: Given a function written in equation form, detect the domain.

1. Identify the input values.
2. Place any restrictions on the input and exclude those values from the domain.
3. Write the domain in interval form, if possible.

Case \(\PageIndex{2}\): Finding the Domain of a Office

Detect the domain of the function \(f(ten)=x^2−one\).

Solution

The input value, shown by the variable x in the equation, is squared and then the result is lowered by one. Whatsoever real number may exist squared and and so be lowered by 1, 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 \((−\infty,\infty)\).

Exercse \(\PageIndex{ii}\)

Detect the domain of the role:

\[f(x)=5−x+x^3 \nonumber\]

Reply

\((−\infty,\infty)\)

Howto: Given a role written in an equation course that includes a fraction, find the domain

1. Identify the input values.
2. Identify whatever restrictions on the input. If at that place is a denominator in the part's formula, fix the denominator equal to zero and solve for ten . If the function's formula contains an even root, prepare the radicand greater than or equal to 0, and and then solve.
3. Write the domain in interval course, making certain to exclude whatever restricted values from the domain.

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

Find the domain of the part \(f(x)=\dfrac{10+1}{2−x}\).

Solution

When there is a denominator, we desire to include only values of the input that do not force the denominator to be cipher. So, we will prepare the denominator equal to 0 and solve for x.

\[ \begin{align*} two−x=0 \\[4pt] −x &=−two \\[4pt] ten&=ii \cease{align*}\]

Now, we will exclude 2 from the domain. The answers are all real numbers where \(x<2\) or \(x>two\). Nosotros can utilize a symbol known every bit the union, \(\cup\),to combine the two sets. In interval annotation, we write the solution:\((−\infty,2)∪(2,\infty)\).

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

Exercse \(\PageIndex{3}\)

Discover the domain of the function:

\[f(x)=\dfrac{1+4x}{2x−1} \nonumber\]

Answer

\[(−\infty,\dfrac{ane}{2})\cup(\dfrac{1}{ii},\infty) \nonumber\]

How To: Given a function written in equation form including an even root, notice the domain.

1. Place the input values.
2. Since there is an even root, exclude any real numbers that upshot in a negative number in the radicand. Fix the radicand greater than or equal to zippo and solve for x.
3. The solution(s) are the domain of the role. If possible, write the answer in interval grade.

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

Find the domain of the function:

\[f(ten)=\sqrt{7-x} \nonumber .\]

Solution

When there is an even root in the formula, we exclude whatsoever existent numbers that result in a negative number in the radicand.

Set the radicand greater than or equal to cypher and solve for x.

\[ \begin{align*} seven−10&≥0 \\[4pt] −ten&≥−7\\[4pt] ten&≤vii \end{align*}\]

Now, we will exclude any number greater than vii from the domain. The answers are all real numbers less than or equal to vii, or \(\left(−\infty,7\correct]\).

Exercse \(\PageIndex{4}\)

Detect the domain of the role

\[f(x)=\sqrt{5+2x}. \nonumber\]

Answer

\[\left[−ii.5,\infty\right) \nonumber\]

Q&A: Tin there be functions in which the domain and range do not intersect at all?

Yes. For example, the part \(f(ten)=-\dfrac{one}{\sqrt{x}}\) has the set up of all positive existent numbers as its domain but the set of all negative real numbers equally its range. As a more than extreme example, a office'south 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 take no elements in mutual.

Using Notations to Specify Domain and Range

In the previous examples, we used inequalities and lists to describe the domain of functions. Nosotros can besides employ inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For instance, \(\{ten|10≤ten<30\}\) describes the behavior of x in set-builder notation. The braces \(\{\}\) are read every bit "the set of," and the vertical bar \(|\) is read as "such that," so we would read\( \{x|10≤x<thirty\}\) as "the set up of x-values such that 10 is less than or equal to 10, and 10 is less than xxx."

Figure \(\PageIndex{4}\) compares inequality notation, prepare-builder notation, and interval notation.

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

To combine two intervals using inequality notation or gear up-builder annotation, we use the word "or." As we saw in earlier examples, we utilise the union symbol, \(\cup\),to combine two unconnected intervals. For example, the union of the sets\(\{2,iii,5\}\) and \(\{4,6\}\) is the set \(\{ii,3,4,5,6\}\). It is the prepare 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 similar these, the elements practise non accept 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 one time in the union gear up. For sets of real numbers on intervals, another instance of a union is

\[\{10| |10|≥3\}=\left(−\infty,−3\correct]\cup\left[3,\infty\right)\]

Ready-Builder Notation and Interval Notation

Set up-builder note is a method of specifying a set of elements that satisfy a certain status. It takes the form\(\{x|\text{ statement about 10}\}\) which is read equally, "the set of all x such that the statement most x is true." For example,

\[\{x|4<10≤12\} \nonumber\]

Interval notation is a mode of describing sets that include all existent numbers between a lower limit that may or may not be included and an upper limit that may or may non 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 note.

1. Identify the intervals to be included in the set by determining where the heavy line overlays the real line.
2. At the left finish of each interval, use [ with each end value to be included in the fix (solid dot) or ( for each excluded stop value (open dot).
3. At the right end of each interval, use ] with each end value to be included in the set up (filled dot) or ) for each excluded end value (open dot).
4. Use the union symbol \(\cup\) to combine all intervals into one gear up.

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

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

Figure \(\PageIndex{five}\): Line graph of \(one≤x≤3 \text{ or }ten>5\).

Solution

To depict the values, \(x\), included in the intervals shown, nosotros would say, "\(x\) is a real number greater than or equal to 1 and less than or equal to three, or a existent number greater than 5."

Inequality

\[one≤ten≤three \text{ or }x>5 \nonumber\]

Set-architect Annotation

\[\{10|1≤x≤iii \text{ or } x>5\}\nonumber\]

Interval notation

\[[1,3]\cup(five,\infty)\nonumber\]

Remember that, when writing or reading interval notation, using a square bracket means the purlieus is included in the set. Using a parenthesis means the purlieus is not included in the set.

Exercse \(\PageIndex{5}\)

Given Figure \(\PageIndex{six}\), specify the graphed fix in

1. words
2. set-builder notation
3. interval notation

Figure \(\PageIndex{six}\): Line graph of \(-2{\leq}x\), \(-i{\leq}x<3\).leq}ten\), \(-one{\leq}x<3\).

Answer a

Values that are less than or equal to –2, or values that are greater than or equal to –1 and less than iii;

Answer b

\(\{10|x≤−2 or −1≤x<3\}\)

Reply c

\(\left(−∞,−two\correct]\cup\left[−1,3\correct)\)

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-centrality. The range is the set of possible output values, which are shown on the y-centrality. Keep in mind that if the graph continues across the portion of the graph nosotros can see, the domain and range may be greater than the visible values. Meet Figure \(\PageIndex{7}\).

Figure \(\PageIndex{vii}\): Graph of a polynomial that shows the 10-centrality is the domain and the y-axis is the range

Nosotros tin observe that the graph extends horizontally from −5 to the correct without bound, so the domain is \(\left[−5,∞\right)\). The vertical extent of the graph is all range values 5 and below, and then the range is \(\left(−∞,v\right]\). Note that the domain and range are ever written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the pinnacle of the graph for range.

Example \(\PageIndex{6A}\): Finding Domain and Range from a Graph

Notice the domain and range of the function f whose graph is shown in Figure 1.two.8.

Figure \(\PageIndex{eight}\): Graph of a function from (-3, 1].

Solution

Nosotros can observe that the horizontal extent of the graph is –3 to 1, and then the domain of f is \(\left(−3,i\right]\).

The vertical extent of the graph is 0 to –4, so the range is \(\left[−4,0\correct)\). Meet Figure \(\PageIndex{ix}\).

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

Case \(\PageIndex{6B}\): 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 \(\PageIndex{ten}\).

Figure \(\PageIndex{10}\): Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per twenty-four hour period and the -centrality is the years(credit: modification of work by the U.S. Energy Information Administration)

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 day," which we 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 \(180≤b≤2010\).

In interval annotation, the domain is \([1973, 2008]\), and the range is nearly \([180, 2010]\). For the domain and the range, we judge the smallest and largest values since they practice not fall exactly on the grid lines.

Exercse \(\PageIndex{6}\)

Given Figure \(\PageIndex{11}\), identify the 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-centrality represents the year.

Answer

domain =\([1950,2002]\)

range = \([47,000,000,89,000,000]\)

Tin a function's domain and range be the same?

Yep. For instance, 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

Nosotros will now return to our set of toolkit functions to decide the domain and range of each.

Figure \(\PageIndex{12}\): Abiding function \(f(ten)=c\).

For the constant office\( f(ten)=c\), the domain consists of all real numbers; at that place are no restrictions on the input. The only output value is the constant \(c\), and so the range is the set \(\{c\}\) that contains this single element. In interval notation, this is written every bit \([c,c]\), the interval that both begins and ends with \(c\).

Figure \(\PageIndex{13}\): Identity function f(10)=ten.

For the identity function \(f(x)=ten\), at that place is no restriction on \(10\). Both the domain and range are the set up of all real numbers.

Effigy \(\PageIndex{xiv}\): Absolute function \(f(x)=|10|\).

For the absolute value function \(f(x)=|ten|\), there is no restriction on \(x\). Even so, considering accented value is divers as a altitude from 0, the output can only be greater than or equal to 0.

Figure \(\PageIndex{15}\): Quadratic office \(f(10)=x^ii\).

For the quadratic role \(f(ten)=ten^ii\), the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Considering the graph does non include any negative values for the range, the range is only nonnegative real numbers.

Figure \(\PageIndex{xvi}\): Cubic office \(f(x)-x^iii\).

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

Effigy \(\PageIndex{17}\): Reciprocal function \(f(x)=\dfrac{ane}{x}\).

For the reciprocal function \(f(x)=\dfrac{1}{10}\), we cannot split past 0, so we must exclude 0 from the domain. Further, i divided by any value can never be 0, so the range as well volition not include 0. In ready-builder note, we could as well write\(\{ten| x≠0\}\),the set of all existent numbers that are non zilch.

Effigy \(\PageIndex{18}\): Reciprocal squared function \(f(x)=\dfrac{1}{ten^two}\)

For the reciprocal squared function \(f(x)=\dfrac{ane}{x^2}\),we cannot split up by 0, and then we must exclude 0 from the domain. There is too no x that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this role is always positive due to the square in the denominator, so the range includes only positive numbers.

Figure \(\PageIndex{nineteen}\): Square root office \(f(x)=\sqrt{(10)}\).

For the square root function \(f(ten)=\sqrt{x}\), we cannot take the square root of a negative real number, and so the domain must exist 0 or greater. The range also excludes negative numbers because the foursquare root of a positive number \(x\) is divers to be positive, even though the foursquare of the negative number \(−\sqrt{10}\) also gives us \(x\).

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

For the cube root function \(f(ten)=\sqrt[3]{ten}\), the domain and range include all real numbers. Note 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 role).

Given the formula for a role, decide the domain and range.

1. Exclude from the domain any input values that effect in division past zero.
2. Exclude from the domain any input values that have nonreal (or undefined) number outputs.
3. Apply the valid input values to make up one's mind the range of the output values.
4. Look at the office graph and table values to ostend 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 and so subtracted from the result.

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

Example \(\PageIndex{7B}\): Finding the Domain and Range

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

Solution

Nosotros cannot evaluate the function at −1 because partition past zero is undefined. The domain is \((−\infty,−i)\loving cup(−1,\infty)\). Because the office is never null, nosotros exclude 0 from the range. The range is \((−\infty,0)\cup(0,\infty)\).

Example \(\PageIndex{7C}\): Finding the Domain and Range

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

Solution

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

\(x+4≥0\) when \(x≥−4\)

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

We and then find the range. We know that \(f(−4)=0\), and the function value increases equally \(ten\) increases without any upper limit. We conclude that the range of f is \(\left[0,\infty\right)\).

Analysis

Figure \(\PageIndex{19}\) represents the office \(f\).

Figure \(\PageIndex{xix}\): Graph of a square root function at \((-4, 0)\).

Exercise \(\PageIndex{seven}\)

Find the domain and range of

\(f(ten)=\sqrt{−2−x}\).

Answer

domain: \(\left(−\infty,-2\right]\)

range: \(\left[0,\infty\correct)\)

Graphing Piecewise-Divers Functions

Sometimes, we see a office that requires more than one formula in lodge to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function \(f(ten)=|ten|\). With a domain of all existent 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 altitude 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.

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

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

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

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

We use piecewise functions to depict situations in which a rule or relationship changes every bit 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. Revenue enhancement brackets are another real-globe case of piecewise functions. For example, consider a simple tax system in which incomes upward to $ten,000 are taxed at 10%, and whatsoever additional income is taxed at 20%. The tax on a total income Southward would be \(0.1S\) if \(South≤$10,000\) and \($k+0.2(Southward−$10,000)\) if \(S>$10,000\).

Piecewise Part

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

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

In piecewise notation, the accented value function is

\[|ten|= \brainstorm{cases} x & \text{if $x \geq 0$} \\ -ten &\text{if $ten<0$} \cease{cases} \nonumber \]

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

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

Instance \(\PageIndex{8A}\): Writing a Piecewise Function

A museum charges $5 per person for a guided bout with a group of 1 to nine people or a fixed $50 fee for a group of x or more than people. Write a part relating the number of people, \(n\), to the cost, \(C\).

Solution

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

\[C(n)= \brainstorm{cases} 5n & \text{if $n < 10$} \\ 50 &\text{if $due north\geq10$} \stop{cases} \nonumber \]

Analysis

The function is represented in Figure \(\PageIndex{20}\). The graph is a diagonal line from \(n=0\) to \(n=x\) and a constant after that. In this instance, the two formulas agree at the meeting point where \(n=x\), only non all piecewise functions have this property.

Figure \(\PageIndex{twenty}\)

Case \(\PageIndex{8B}\): Working with a Piecewise Function

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

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

Find the cost of using i.5 gigabytes of data and the cost of using iv gigabytes of data.

Soltuion

To find the price of using 1.five gigabytes of information, \(C(1.5)\), nosotros first await to meet which part of the domain our input falls in. Because 1.5 is less than ii, we utilise the offset formula.

\[C(i.5)=$25 \nonumber \]

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

\[C(4)=25+10(iv−2)=$45 \nonumber \]

Analysis

The function is represented in Figure \(\PageIndex{21}\). Nosotros can see where the function changes from a constant to a shifted and stretched identity at \(one thousand=two\). Nosotros plot the graphs for the different formulas on a common prepare of axes, making certain each formula is practical on its proper domain.

Figure \(\PageIndex{21}\)

Given a piecewise function, sketch a graph.

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

Example \(\PageIndex{8C}\): Graphing a Piecewise Part

Sketch a graph of the part.

\[f(x)= \begin{cases} x^two & \text{if $x \leq 1$} \\ 3 &\text{if $one<x\leq2$} \\ 10 &\text{if $10>ii$} \end{cases} \nonumber \]

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 so limiting the graph to the indicated domain. At the endpoints of the domain, we describe open circles to signal where the endpoint is not included considering of a less-than or greater-than inequality; we draw a airtight 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 divide coordinate systems.

Effigy \(\PageIndex{20}\): Graph of each part of the slice-wise function f(10)

(a)\( f(10)=x^2\) if \(x≤i\); (b) \(f(x)=3\) if \(1< ten≤ii\); (c) \(f(x)=10\) if \(x>2\)

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

Effigy \(\PageIndex{21}\): Graph of the entire role.

Assay

Note that the graph does pass the vertical line test even at \(x=1\) and \(x=2\) because the points \((1,3)\) and \((two,2)\) are non part of the graph of the office, though \((one,i)\) and \((2, 3)\) are.

Exercise \(\PageIndex{viii}\)

Graph the post-obit piecewise role.

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

Respond

Effigy \(\PageIndex{22}\)

Can more than 1 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 naught 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 up of ordered pairs.
• The domain of a function can too be determined by identifying the input values of a part written every bit an equation.
• Interval values represented on a number line tin can exist described using inequality notation, gear up-builder note, and interval notation.
• For many functions, the domain and range can exist determined from a graph.
• An agreement of toolkit functions can exist used to observe the domain and range of related functions.
• A piecewise office is described past more than one formula.
• A piecewise function can be graphed using each algebraic formula on its assigned subdomain.

Glossary

interval notation

a method of describing a set 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 up, and a parenthesis indicating exclusion

piecewise function

a function in which more than one formula is used to define the output

set-builder note

a method of describing a set by a dominion that all of its members obey; it takes the class {x| statement about x}

Domain And Range Piecewise Functions,

Source: https://math.libretexts.org/Sandboxes/Jaison_Novick/1250_Draft_4/02%3A_Functions/2.03%3A_Domain_and_Range_Piecewise_Functions

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