Chapter 1 Cumulative Review Identify the Domain and Range

LEARNING OBJECTIVES

Past the end of this lesson, you will be able to:

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

If you lot're in the mood for a scary movie, yous may want to check out one of the 5 most popular horror movies of all time—I am Legend, Hannibal, The Ring, The Grudge, and The Conjuring. Effigy 1 shows the amount, in dollars, each of those movies grossed when they were released too every bit the ticket sales for horror movies in full general by year. Notice that we can employ the data to create a function of the amount each movie earned or the total ticket sales for all horror movies past year. In creating various functions using the data, we tin can identify dissimilar independent and dependent variables, and we can analyze the data and the functions to make up one's mind the domain and range. In this section, we volition investigate methods for determining the domain and range of functions such as these.

Discover the domain of a function divers by an equation

In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section, we will exercise determining domains and ranges for specific functions. Proceed in mind that, in determining domains and ranges, nosotros need to consider what is physically possible or meaningful in real-globe examples, such equally tickets sales and twelvemonth in the horror pic example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of existent numbers. Or in a function expressed every bit a formula, nosotros cannot include any input value in the domain that would lead us to divide by 0.

Figure two

We can visualize the domain equally a "holding area" that contains "raw materials" for a "function machine" and the range as another "belongings surface area" for the motorcar'south products.

We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval annotation, we utilise a foursquare bracket [ when the ready includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, he or she would demand to express the interval that is more than 0 and less than or equal to 100 and write [latex]\left(0,\text{ }100\right][/latex]. Nosotros will discuss interval notation in greater detail later.

Let's turn our attending to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering 3 different forms. Starting time, if the role has no denominator or an even root, consider whether the domain could be all existent numbers. Second, if there is a denominator in the function's equation, exclude values in the domain that strength the denominator to be zilch. 3rd, if at that place is an fifty-fifty root, consider excluding values that would brand the radicand negative.

Before we begin, permit united states review the conventions of interval note:

• The smallest term from the interval is written outset.
• The largest term in the interval is written second, following a comma.
• Parentheses, ( or ), are used to signify that an endpoint is not included, chosen exclusive.
• Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.

The table beneath gives a summary of interval notation.

Example 1: Finding the Domain of a Function equally a Prepare of Ordered Pairs

Find the domain of the following function: [latex]\left\{\left(2,\text{ }10\right),\left(3,\text{ }10\right),\left(iv,\text{ }20\right),\left(5,\text{ }30\correct),\left(6,\text{ }twoscore\correct)\right\}[/latex] .

Solution

First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, every bit the ordered pairs are merely listed. The domain is the set up of the get-go coordinates of the ordered pairs.

[latex]\left\{2,3,iv,5,half dozen\right\}[/latex]

Try Information technology 1

Notice the domain of the function:

[latex]\left\{\left(-5,4\right),\left(0,0\right),\left(5,-4\right),\left(10,-8\right),\left(xv,-12\right)\right\}[/latex]

Solution

How To: Given a office written in equation form, observe the domain.

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

Example two: Finding the Domain of a Role

Observe the domain of the office [latex]f\left(x\right)={ten}^{2}-1[/latex].

Solution

The input value, shown by the variable [latex]x[/latex] in the equation, is squared and so the result is lowered by 1. Whatever existent number may be squared and then be lowered by one, so there are no restrictions on the domain of this office. The domain is the set of real numbers.

In interval form, the domain of [latex]f[/latex] is [latex]\left(-\infty ,\infty \right)[/latex].

Endeavour It 2

Observe the domain of the function: [latex]f\left(10\right)=5-x+{x}^{3}[/latex].

Solution

How To: Given a role written in an equation form that includes a fraction, find the domain.

1. Identify the input values.
2. Identify any restrictions on the input. If in that location is a denominator in the office's formula, set up the denominator equal to goose egg and solve for [latex]10[/latex] . If the role'due south formula contains an even root, set the radicand greater than or equal to 0, then solve.
3. Write the domain in interval form, making sure to exclude whatsoever restricted values from the domain.

Example 3: Finding the Domain of a Function Involving a Denominator (Rational Function)

Find the domain of the part [latex]f\left(x\correct)=\frac{x+i}{two-10}[/latex].

Solution

When at that place is a denominator, we want to include only values of the input that do not force the denominator to be zero. And then, we volition set up the denominator equal to 0 and solve for [latex]x[/latex].

[latex]\begin{cases}2-ten=0\hfill \\ -x=-2\hfill \\ x=2\hfill \stop{cases}[/latex]

Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x<2[/latex] or [latex]x>ii[/latex]. We tin can utilize a symbol known equally the union, [latex]\loving cup [/latex], to combine the ii sets. In interval notation, we write the solution: [latex]\left(\mathrm{-\infty },2\right)\cup \left(2,\infty \right)[/latex].

Figure iii

In interval form, the domain of [latex]f[/latex] is [latex]\left(-\infty ,2\right)\cup \left(two,\infty \right)[/latex].

Endeavour It iii

Notice the domain of the function: [latex]f\left(x\right)=\frac{1+4x}{2x - ane}[/latex].

Solution

How To: Given a function written in equation grade including an fifty-fifty root, notice the domain.

1. Identify the input values.
2. Since there is an fifty-fifty root, exclude whatsoever real numbers that event in a negative number in the radicand. Set the radicand greater than or equal to zip and solve for [latex]10[/latex].
3. The solution(s) are the domain of the function. If possible, write the answer in interval course.

Instance 4: Finding the Domain of a Function with an Even Root

Find the domain of the function [latex]f\left(x\correct)=\sqrt{vii-x}[/latex].

Solution

When there is an fifty-fifty root in the formula, nosotros 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 [latex]x[/latex].

[latex]\brainstorm{cases}seven-ten\ge 0\hfill \\ -10\ge -7\hfill \\ ten\le seven\hfill \end{cases}[/latex]

Now, we will exclude any number greater than seven from the domain. The answers are all existent numbers less than or equal to [latex]7[/latex], or [latex]\left(-\infty ,7\correct][/latex].

Endeavour It iv

Find the domain of the function [latex]f\left(x\right)=\sqrt{5+2x}[/latex].

Solution

Q & A

Can in that location be functions in which the domain and range do not intersect at all?

Yes. For case, the function [latex]f\left(10\right)=-\frac{1}{\sqrt{10}}[/latex] has the ready of all positive real numbers as its domain only the prepare of all negative real numbers every bit 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 have no elements in mutual.

In the previous examples, nosotros used inequalities and lists to depict the domain of functions. We can as well use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set up-builder annotation. For instance, [latex]\left\{x|10\le x<30\right\}[/latex] describes the behavior of [latex]ten[/latex] 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 [latex]\left\{x|10\le x<30\right\}[/latex] as "the fix of x-values such that 10 is less than or equal to [latex]x[/latex], and [latex]10[/latex] is less than 30."

The table below compares inequality notation, set-architect annotation, and interval annotation.

	Inequality Notation	Set-builder Note	Interval Note
	5 < h ≤ 10	{ h | 5 < h ≤ 10}	(5, x]
	v ≤ h < ten	{ h | five ≤ h < 10}	[5, 10]
	v < h < 10	{ h | 5 < 10 }	(v, 10)
	h < 10	{ h | h < 10 }	( −∞, 10)
	h ≥ ten	{ h | h ≥ 10 }	[10, ∞ )
	All real numbers	ℝ	( −∞, ∞ )

To combine ii intervals using inequality notation or ready-builder notation, we apply the word "or." As we saw in earlier examples, we use the union symbol, [latex]\cup [/latex], to combine two unconnected intervals. For case, the union of the sets [latex]\left\{2,three,5\right\}[/latex] and [latex]\left\{iv,half dozen\correct\}[/latex] is the set [latex]\left\{2,3,4,five,6\correct\}[/latex]. It is the set up 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 non take to be listed in ascending order of numerical value. If the original 2 sets have some elements in common, those elements should be listed only one time in the union set. For sets of real numbers on intervals, another case of a union is

[latex]\left\{x|\text{ }|ten|\ge 3\right\}=\left(-\infty ,-3\right]\cup \left[3,\infty \right)[/latex]

This video describes how to use interval note to describe a ready.

This video describes how to use Ready-Architect note to describe a set.

A Full general Note: Ready-Architect Notation and Interval Notation

Set-builder notation is a method of specifying a set of elements that satisfy a sure condition. It takes the course [latex]\left\{10|\text{statement almost }x\right\}[/latex] which is read as, "the set of all [latex]x[/latex] such that the statement almost [latex]x[/latex] is true." For instance,

[latex]\left\{x|4<x\le 12\correct\}[/latex]

Interval annotation is a fashion of describing sets that include all real numbers betwixt 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 betwixt brackets or parentheses. A square subclass indicates inclusion in the set, and a parenthesis indicates exclusion from the prepare. For example,

[latex]\left(4,12\right][/latex]

How To: Given a line graph, describe the ready of values using interval note.

1. Identify the intervals to exist included in the fix by determining where the heavy line overlays the existent line.
2. At the left terminate of each interval, employ [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open up dot).
3. At the right end of each interval, apply ] with each cease value to be included in the set (filled dot) or ) for each excluded end value (open dot).
4. Use the wedlock symbol [latex]\cup [/latex] to combine all intervals into one gear up.

Example v: Describing Sets on the Existent-Number Line

Describe the intervals of values shown in Figure 4 using inequality notation, set-builder notation, and interval notation.

Figure four

Solution

To draw the values, [latex]x[/latex], included in the intervals shown, we would say, " [latex]x[/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5."

Inequality	[latex]1\le x\le iii\text{or}x>5[/latex]
Fix-builder notation	[latex]\left\{10|1\le x\le 3\text{or}x>5\right\}[/latex]
Interval notation	[latex]\left[i,3\right]\cup \left(5,\infty \right)[/latex]

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

Try Information technology 5

Given Figure v, specify the graphed gear up in

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

Effigy v

Solution

Finding Domain and Range from Graphs

Another way to identify the domain and range of functions is past 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 ready of possible output values, which are shown on the y-axis. Proceed in mind that if the graph continues beyond the portion of the graph we can run into, the domain and range may be greater than the visible values. See Figure half dozen.

Effigy vi

We tin can detect that the graph extends horizontally from [latex]-five[/latex] to the right without bound, so the domain is [latex]\left[-5,\infty \right)[/latex]. The vertical extent of the graph is all range values [latex]5[/latex] and below, so the range is [latex]\left(\mathrm{-\infty },5\right][/latex]. Annotation 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 meridian of the graph for range.

Example 6: Finding Domain and Range from a Graph

Find the domain and range of the function [latex]f[/latex] whose graph is shown in Figure 7.

Figure vii

Solution

We tin detect that the horizontal extent of the graph is –three to 1, so the domain of [latex]f[/latex] is [latex]\left(-iii,1\right][/latex].

Figure 8

The vertical extent of the graph is 0 to –4, so the range is [latex]\left[-4,0\right)[/latex].

Example 7: Finding Domain and Range from a Graph of Oil Production

Observe the domain and range of the function [latex]f[/latex] whose graph is shown in Effigy 9.

Solution

The input quantity along the horizontal axis is "years," which we represent with the variable [latex]t[/latex] for fourth dimension. The output quantity is "thousands of barrels of oil per day," which we represent with the variable [latex]b[/latex] for barrels. The graph may continue to the left and correct beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain every bit [latex]1973\le t\le 2008[/latex] and the range as approximately [latex]180\le b\le 2010[/latex].

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 practice not autumn exactly on the grid lines.

Try It vi

Given the graph in Figure ten, identify the domain and range using interval notation.

Figure 10

Solution

Q & A

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

Yes. For example, the domain and range of the cube root role are both the gear up of all real numbers.

Finding Domain and Range from Graphs

Nosotros volition now return to our set up of toolkit functions to determine the domain and range of each.

11

Figure 11. For the constant function [latex]f\left(x\right)=c[/latex], the domain consists of all real numbers; there are no restrictions on the input. The merely output value is the constant [latex]c[/latex], so the range is the set [latex]\left\{c\right\}[/latex] that contains this single element. In interval notation, this is written as [latex]\left[c,c\right][/latex], the interval that both begins and ends with [latex]c[/latex].

12

Effigy 12. For the identity function [latex]f\left(x\right)=x[/latex], there is no brake on [latex]ten[/latex]. Both the domain and range are the set up of all real numbers.

13

Effigy thirteen. For the accented value part [latex]f\left(x\right)=|x|[/latex], there is no restriction on [latex]x[/latex]. However, because absolute value is divers equally a altitude from 0, the output can merely be greater than or equal to 0.

14

Figure 14. For the quadratic role [latex]f\left(x\right)={x}^{2}[/latex], 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 whatsoever negative values for the range, the range is merely nonnegative real numbers.

15

Figure xv. For the cubic role [latex]f\left(10\right)={10}^{3}[/latex], 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 existent numbers.

16

Figure sixteen. For the reciprocal function [latex]f\left(ten\right)=\frac{i}{x}[/latex], nosotros cannot divide past 0, and then we must exclude 0 from the domain. Further, 1 divided past whatever value can never be 0, so the range also volition non include 0. In set-builder annotation, we could also write [latex]\left\{x|\text{ }x\ne 0\right\}[/latex], the set of all real numbers that are not naught.

17

Figure 17. For the reciprocal squared office [latex]f\left(10\right)=\frac{1}{{ten}^{2}}[/latex], we cannot carve up past [latex]0[/latex], so nosotros must exclude [latex]0[/latex] from the domain. There is also no [latex]x[/latex] that can give an output of 0, and then 0 is excluded from the range likewise. Note that the output of this function is ever positive due to the foursquare in the denominator, so the range includes only positive numbers.

xviii

Figure 18. For the square root function [latex]f\left(ten\right)=\sqrt[]{x}[/latex], we cannot accept the foursquare root of a negative existent number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number [latex]x[/latex] is divers to exist positive, even though the square of the negative number [latex]-\sqrt{x}[/latex] also gives the states [latex]x[/latex].

nineteen

Figure 19. For the cube root function [latex]f\left(x\right)=\sqrt[3]{x}[/latex], the domain and range include all real numbers. Annotation that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (information technology is an odd function).

How To: Given the formula for a role, determine the domain and range.

1. Exclude from the domain any input values that result in segmentation past nothing.
2. Exclude from the domain whatsoever input values that take nonreal (or undefined) number outputs.
3. Apply the valid input values to determine the range of the output values.
4. Await at the role graph and table values to confirm the actual function behavior.

Case 8: Finding the Domain and Range Using Toolkit Functions

Observe the domain and range of [latex]f\left(x\right)=ii{ten}^{three}-x[/latex].

Solution

In that location are no restrictions on the domain, as whatever real number may be cubed and so subtracted from the upshot.

The domain is [latex]\left(-\infty ,\infty \correct)[/latex] and the range is also [latex]\left(-\infty ,\infty \right)[/latex].

Case 9: Finding the Domain and Range

Find the domain and range of [latex]f\left(10\right)=\frac{2}{x+i}[/latex].

Solution

We cannot evaluate the function at [latex]-i[/latex] because sectionalization by nada is undefined. The domain is [latex]\left(-\infty ,-i\correct)\cup \left(-ane,\infty \right)[/latex]. Considering the role is never zero, nosotros exclude 0 from the range. The range is [latex]\left(-\infty ,0\correct)\loving cup \left(0,\infty \correct)[/latex].

Example 10: Finding the Domain and Range

Notice the domain and range of [latex]f\left(x\right)=2\sqrt{x+4}[/latex].

Solution

We cannot take the square root of a negative number, and then the value inside the radical must exist nonnegative.

[latex]10+4\ge 0\text{ when }10\ge -4[/latex]

The domain of [latex]f\left(x\correct)[/latex] is [latex]\left[-4,\infty \right)[/latex].

We then detect the range. We know that [latex]f\left(-iv\correct)=0[/latex], and the function value increases as [latex]ten[/latex] increases without any upper limit. We conclude that the range of [latex]f[/latex] is [latex]\left[0,\infty \correct)[/latex].

Analysis of the Solution

Figure 20 represents the function [latex]f[/latex].

Figure 20

Endeavor It 7

Notice the domain and range of [latex]f\left(x\correct)=-\sqrt{ii-x}[/latex].

Solution

Graphic Piecewise-Defined Functions

Sometimes, we run into a role that requires more than one formula in lodge to obtain the given output. For example, in the toolkit functions, we introduced the absolute value role [latex]f\left(x\right)=|10|[/latex]. With a domain of all real numbers and a range of values greater than or equal to 0, accented 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 crave the output to exist greater than or equal to 0.

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

[latex]f\left(x\correct)=x\text{ if }x\ge 0[/latex]

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

[latex]f\left(10\right)=-x\text{ if }x<0[/latex]

Because this requires two dissimilar processes or pieces, the accented value function is an example of a piecewise function. A piecewise office is a function in which more than than one formula is used to ascertain the output over unlike pieces of the domain.

We apply piecewise functions to draw situations in which a rule or relationship changes as the input value crosses sure "boundaries." For example, we often encounter situations in business for which the price per piece of a sure particular is discounted once the number ordered exceeds a certain value. Revenue enhancement brackets are another existent-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and whatsoever additional income is taxed at xx%. The tax on a full income, S, would be 0.1S if [latex]{S}\le\[/latex] $10,000 and yard + 0.2 (S – $10,000), if Due south> $10,000.

A Full general Note: Piecewise Role

A piecewise function is a part in which more than 1 formula is used to ascertain the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:

[latex] f\left(ten\right)=\brainstorm{cases}\text{formula 1 if x is in domain 1}\\ \text{formula ii if ten is in domain 2}\\ \text{formula 3 if 10 is in domain 3}\end{cases} [/latex]

In piecewise notation, the absolute value function is

[latex]|10|=\begin{cases}x\text{ if }x\ge 0\\ -x\text{ if }x<0\cease{cases}[/latex]

How To: Given a piecewise role, write the formula and place the domain for each interval.

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

Instance xi: Writing a Piecewise Role

A museum charges $5 per person for a guided tour with a group of one to ix people or a stock-still $50 fee for a group of 10 or more people. Write a part relating the number of people, [latex]due north[/latex], to the cost, [latex]C[/latex].

Solution

Two different formulas will be needed. For northward-values under 10, C=5n. For values of n that are 10 or greater, C=l.

C(n)=[latex]\brainstorm{cases}{5n}\text{ if }{0}<{n}<{10}\\ 50\text{ if }{n}\ge 10\end{cases}[/latex]

Analysis of the Solution

The function is represented in Figure 21. The graph is a diagonal line from [latex]n=0[/latex] to [latex]northward=10[/latex] and a constant afterward that. In this case, the two formulas concur at the meeting point where [latex]n=10[/latex], just non all piecewise functions accept this property.

Effigy 21

Case 12: Working with a Piecewise Role

A cell telephone company uses the office below to determine the cost, [latex]C[/latex], in dollars for [latex]grand[/latex] gigabytes of data transfer.

[latex]C\left(thou\right)=\begin{cases}{25} \text{ if }{ 0 }<{ g }<{ 2 }\\ { 25+10 }\left(thou - ii\correct) \text{ if }{ g}\ge{ 2 }\finish{cases}[/latex]

Find the cost of using 1.v gigabytes of data and the price of using 4 gigabytes of information.

Solution

To find the price of using 1.5 gigabytes of data, C(ane.v), nosotros first expect to see which role of the domain our input falls in. Because i.5 is less than 2, we use the first formula.

C(1.5) = $25

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

C(iv)=25 + x( four-two) =$45

How To: Given a piecewise function, sketch a graph.

1. Signal on the x-centrality the boundaries defined 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 non graph two functions over one interval because it would violate the criteria of a function.

Example 13: Graphing a Piecewise Function

Sketch a graph of the part.

[latex]f\left(x\right)=\brainstorm{cases}{ x }^{2} \text{ if }{ ten }\le{ i }\\ { 3 } \text{ if } { i }<{ x }\le 2\\ { x } \text{ if }{ 10 }>{ 2 }\end{cases}[/latex]

Solution

Each of the component functions is from our library of toolkit functions, and then 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, nosotros draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a airtight circle where the endpoint is included considering of a less-than-or-equal-to or greater-than-or-equal-to inequality.

Below are the three components of the piecewise part graphed on carve up coordinate systems.

(a) [latex]f\left(ten\right)={x}^{2}\text{ if }ten\le 1[/latex]; (b) [latex]f\left(x\correct)=3\text{ if i< }10\le ii[/latex]; (c) [latex]f\left(x\right)=x\text{ if }x>ii[/latex]

Effigy 23

At present that nosotros take sketched each piece individually, we combine them in the same coordinate plane.

Figure 24

Endeavor It 8

Graph the following piecewise part.

[latex]f\left(ten\right)=\begin{cases}{ ten}^{3} \text{ if }{ x }<{-1 }\\ { -2 } \text{ if } { -1 }<{ 10 }<{ 4 }\\ \sqrt{x} \text{ if }{ x }>{ 4 }\end{cases}[/latex]

Q&A

Tin more than one formula from a piecewise part 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 role includes all real input values that would not cause united states of america to attempt an undefined mathematical performance, such as dividing by cypher 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 past identifying the input values of a part written as an equation.
• Interval values represented on a number line tin be described using inequality notation, ready-builder notation, and interval notation.
• For many functions, the domain and range can be adamant 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 tin exist graphed using each algebraic formula on its assigned subdomain.

1. Why does the domain differ for dissimilar functions?
2. How do we decide the domain of a function defined by an equation?
3. Explicate why the domain of [latex]f\left(x\correct)=\sqrt[3]{x}[/latex] is unlike from the domain of [latex]f\left(ten\correct)=\sqrt[]{x}[/latex].
4. When describing sets of numbers using interval notation, when do you lot apply a parenthesis and when exercise you lot use a subclass?
5. How do you lot graph a piecewise function?

For the following exercises, find the domain of each office using interval notation.

half dozen. [latex]f\left(10\correct)=-2x\left(10 - 1\right)\left(x - two\right)[/latex]

vii. [latex]f\left(x\right)=5 - 2{x}^{2}[/latex]

8. [latex]f\left(x\correct)=3\sqrt{10 - 2}[/latex]

nine. [latex]f\left(10\right)=iii-\sqrt{half-dozen - 2x}[/latex]

10. [latex]f\left(ten\correct)=\sqrt{iv - 3x}[/latex]

11. [latex]f\left(10\correct)=\sqrt{{x}^{2}+4}[/latex]

12. [latex]f\left(x\right)=\sqrt[3]{one - 2x}[/latex]

13. [latex]f\left(x\right)=\sqrt[3]{ten - ane}[/latex]

14. [latex]f\left(x\right)=\frac{9}{x - six} \\ [/latex]

15. [latex]f\left(x\correct)=\frac{3x+1}{4x+ii} [/latex]

sixteen. [latex]f\left(x\right)=\frac{\sqrt{x+iv}}{ten - 4} [/latex]

17. [latex]f\left(x\correct)=\frac{x - 3}{{x}^{2}+9x - 22} [/latex]

eighteen. [latex]f\left(ten\correct)=\frac{ane}{{10}^{2}-x - six} [/latex]

19. [latex]f\left(ten\right)=\frac{2{10}^{3}-250}{{x}^{ii}-2x - 15} [/latex]

twenty. [latex]\frac{5}{\sqrt{x - 3}} [/latex]

21. [latex]\frac{2x+one}{\sqrt{5-x}} [/latex]

22. [latex]f\left(10\right)=\frac{\sqrt{x - iv}}{\sqrt{x - 6}} [/latex]

23. [latex]f\left(x\right)=\frac{\sqrt{ten - 6}}{\sqrt{x - iv}} [/latex]

24. [latex]f\left(x\right)=\frac{x}{x} [/latex]

25. [latex]f\left(x\right)=\frac{{x}^{2}-9x}{{ten}^{two}-81} [/latex]

26. Notice the domain of the function [latex]f\left(x\right)=\sqrt{ii{10}^{3}-50x} [/latex] by:

a. using algebra.
b. graphing the office in the radicand and determining intervals on the x-axis for which the radicand is nonnegative.

For the post-obit exercises, write the domain and range of each function using interval notation.

27.

Domain: ________ Range: ________

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

38. [latex]f(x)=\begin{cases}{ten}+{1}&\text{ if }&{ x }<{ -2 } \\{-2x - 3}&\text{ if }&{ x }\ge { -2 }\\ \end{cases} [/latex]

39. [latex]f\left(x\right)=\begin{cases}{2x - one}&\text{ if }&{ ten }<{ 1 }\\ {1+x }&\text{ if }&{ ten }\ge{ one } \end{cases}[/latex]

40. [latex]f\left(x\correct)=\brainstorm{cases}{x+one}&\text{ if }&{ ten }<{ 0 }\\ {10 - 1 }&\text{ if }&{ x }>{ 0 }\cease{cases}[/latex]

41. [latex]f\left(x\right)=\brainstorm{cases}{3} &\text{ if }&{ ten } <{ 0 }\\ \sqrt{x}&\text{ if }&{ x }\ge { 0 }\end{cases}[/latex]

42. [latex]f\left(ten\right)=\begin{cases}{ten}^{2}&\text{ if }&{ ten } <{ 0 }\\ {1-x}&\text{ if }&{ x } >{ 0 }\cease{cases}[/latex]

43. [latex]f\left(x\right)=\begin{cases}{x}^{2}&\text{ if }&{ ten }<{ 0 }\\ {x+2 }&\text{ if }&{ x }\ge { 0 }\end{cases}[/latex]

44. [latex]f\left(x\right)=\brainstorm{cases}x+1& \text{if}& ten<ane\\ {x}^{iii}& \text{if}& x\ge one\terminate{cases}[/latex]

45. [latex]f\left(x\right)=\begin{cases}|x|&\text{ if }&{ 10 }<{ 2 }\\ { 1 }&\text{ if }&{ x }\ge{ 2 }\end{cases}[/latex]
For the following exercises, given each function [latex]f[/latex], evaluate [latex]f\left(-3\correct),f\left(-2\right),f\left(-i\right)[/latex], and [latex]f\left(0\correct)[/latex].

46. [latex]f\left(x\correct)=\begin{cases}{ x+ane }&\text{ if }&{ 10 }<{ -2 }\\ { -2x - 3 }&\text{ if }&{ ten }\ge{ -two }\end{cases}[/latex]

47. [latex]f\left(ten\right)=\begin{cases}{ 1 }&\text{ if }&{ x }\le{ -3 }\\{ 0 }&\text{ if }&{ x }>{ -3 }\cease{cases}[/latex]

48. [latex]f\left(x\right)=\begin{cases}{-2}{x}^{two}+{ three }&\text{ if }&{ x }\le { -ane }\\ { 5x } - { 7 } &\text{ if }&{ x } > { -1 }\finish{cases}[/latex]
For the following exercises, given each function [latex]f[/latex], evaluate [latex]f\left(-one\right),f\left(0\right),f\left(2\right)[/latex], and [latex]f\left(4\right)[/latex].

49. [latex]f\left(x\correct)=\begin{cases}{ 7x+3 }&\text{ if }&{ x }<{ 0 }\\{ 7x+6 }&\text{ if }&{ x }\ge{ 0 }\cease{cases}[/latex]

50. [latex]f\left(x\right)=\begin{cases}{x}^{ii}{ -2 }&\text{ if }&{ ten }<{ 2 }\\{ 4+|x - 5|}&\text{ if }&{ x }\ge{ 2 }\finish{cases}[/latex]

51. [latex]f\left(ten\right)=\begin{cases}5x& \text{if}& x<0\\ 3& \text{if}& 0\le ten\le 3\\ {10}^{2}& \text{if}& x>3\cease{cases}[/latex]
For the following exercises, write the domain for the piecewise function in interval notation.

52. [latex]f\left(x\right)=\begin{cases}{x+1}&\text{ if }&{ x }<{ -2 }\\{ -2x - iii}&\text{ if }&{ 10 }\ge{ -two }\end{cases}[/latex]

53. [latex]f\left(x\correct)=\brainstorm{cases}{x}^{ii}{ -2 }&\text{ if}&{ ten }<{ 1 }\\{-x}^{two}+{2}&\text{ if }&{ x }>{ 1 }\cease{cases}[/latex]

54. [latex]f\left(x\right)=\begin{cases}{ 2x - three }&\text{ if }&{ x }<{ 0 }\\{ -3}{x}^{2}&\text{ if }&{ 10 }\ge{ 2 }\finish{cases}[/latex]

55. Graph [latex]y=\frac{1}{{10}^{2}}[/latex] on the viewing window [latex]\left[-0.5,-0.1\right][/latex] and [latex]\left[0.1,0.5\right][/latex]. Determine the corresponding range for the viewing window. Show the graphs.

56. Graph [latex]y=\frac{one}{x}[/latex] on the viewing window [latex]\left[-0.5,-0.one\right][/latex] and [latex]\left[0.one,\text{ }0.v\right][/latex]. Determine the respective range for the viewing window. Show the graphs.

57. Suppose the range of a function [latex]f[/latex] is [latex]\left[-5,\text{ }8\correct][/latex]. What is the range of [latex]|f\left(x\right)|?[/latex]

58. Create a office in which the range is all nonnegative real numbers.

59 .Create a role in which the domain is [latex]x>2[/latex].

threescore. The cost in dollars of making [latex]10[/latex] items is given by the role [latex]C\left(x\right)=10x+500[/latex].

A. The fixed cost is determined when zero items are produced. Find the fixed price for this item.
B. What is the cost of making 25 items?
C. Suppose the maximum price allowed is $1500. What are the domain and range of the cost office, [latex]C\left(10\correct)?[/latex]

61. The height [latex]h[/latex] of a projectile is a function of the time [latex]t[/latex] it is in the air. The height in feet for [latex]t[/latex] seconds is given past the function [latex]h\left(t\right)=-16{t}^{2}+96t[/latex]. What is the domain of the function? What does the domain hateful in the context of the problem?

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 subclass indicating inclusion in the set, and a parenthesis indicating exclusion

piecewise part

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

ready-builder notation

a method of describing a set by a rule that all of its members obey; it takes the form [latex]\left\{x|\text{argument virtually }x\correct\}[/latex]

kuhnwomers1988.blogspot.com

Source: https://courses.lumenlearning.com/precalcone/chapter/domain-and-range/

Share this post