Domain and Range

Finding the Domain of a Function Defined by an Equation

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

We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See .

We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write \((0,\ 100].\) We will discuss interval notation in greater detail later.

Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.

Before we begin, let us review the conventions of interval notation:

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

See for a summary of interval notation.

Condensed — the full section is in OpenStax Precalculus 2e.

Using Notations to Specify Domain and Range

In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example, \(\{x|10\le x<30\}\) describes the behavior of \(x\) in set-builder notation. The braces \(\{\}\) are read as “the set of,” and the vertical bar | is read as “such that,” so we would read \(\{x|10\le x<30\}\) as “the set of x-values such that 10 is less than or equal to \(x,\) and \(x\) is less than 30.”

compares inequality notation, set-builder notation, and interval notation.

To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol, \(\cup ,\) to combine two unconnected intervals. For example, the union of the sets \(\{2,3,5\}\) and \(\{4,6\}\) is the set \(\{2,3,4,5,6\}.\) It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

\[\{x|\ |x|\ge 3\}=(-\infty ,-3]\cup [3,\infty )\]

Condensed — the full section is in OpenStax Precalculus 2e.

Finding Domain and Range from Graphs

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See .

We can observe that the graph extends horizontally from \(-5\) to the right without bound, so the domain is \([-5,\infty ).\) The vertical extent of the graph is all range values \(5\) and below, so the range is \((-\infty ,5].\) Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.

Example

Try it.

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

Solution

We can observe that the horizontal extent of the graph is –3 to 1, so the domain of \(f\) is \((-3,1].\)

The vertical extent of the graph is 0 to –4, so the range is \([-4,0].\) See .

Example

Try it.

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

Solution

The input quantity along the horizontal axis is “years,” which we represent with the variable \(t\) for time. The output quantity is “thousands of barrels of oil per day,” which we represent with the variable \(b\) for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as \(1973\le t\le 2008\) and the range as approximately \(180\le b\le 2010.\)

In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

Finding Domains and Ranges of the Toolkit Functions

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

Example

Try it.

Find the domain and range of \(f(x)=2{x}^{3}-x.\)

Solution

There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

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

Example

Try it.

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

Solution

We cannot evaluate the function at \(-1\) because division by zero is undefined. The domain is \((-\infty ,-1)\cup (-1,\infty ).\) Because the function is never zero, we exclude 0 from the range. The range is \((-\infty ,0)\cup (0,\infty ).\)

Example

Try it.

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

Solution

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

\[x+4\ge 0\text{ when }x\ge -4\]

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

We then find the range. We know that \(f(-4)=0,\) and the function value increases as \(x\) increases without any upper limit. We conclude that the range of \(f\) is \([0,\infty ).\)

Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function \(f(x)=|x|.\) With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude, or modulus, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

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

\[f(x)=x\ \text{if}\ x\ge 0\]

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

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

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

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income \(S\) would be \(0.1S\) if \(S\le \text{\$}10\text{,}000\) and \(\text{\$}1000+0.2(S-\text{\$}10\text{,}000)\) if \(S>\text{\$}10\text{,}000.\)

Condensed — the full section is in OpenStax Precalculus 2e.

Key Concepts

• The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.
• The domain of a function can be determined by listing the input values of a set of ordered pairs. See .
• The domain of a function can also be determined by identifying the input values of a function written as an equation. See , , and .
• Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See .
• For many functions, the domain and range can be determined from a graph. See and .
• An understanding of toolkit functions can be used to find the domain and range of related functions. See , , and .
• A piecewise function is described by more than one formula. See and .
• A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See .

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Find the domain of the following function: \(\{(2,\ 10),(3,\ 10),(4,\ 20),(5,\ 30),(6,\ 40)\}\).

   အဖြေကို ဖော်ပြပါ

   First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.

   \[\{2,3,4,5,6\}\]

2. Find the domain of the function:

   \(\{(-5,4),(0,0),(5,-4),(10,-8),(15,-12)\}\)

   အဖြေကို ဖော်ပြပါ

   \(\{-5,\ 0,\ 5,\ 10,\ 15\}\)

3. Find the domain of the function \(f(x)={x}^{2}-1.\)

   အဖြေကို ဖော်ပြပါ

   The input value, shown by the variable \(x\) in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.

   In interval form, the domain of \(f\) is \((-\infty ,\infty ).\)

4. Find the domain of the function: \(f(x)=5-x+{x}^{3}.\)

   အဖြေကို ဖော်ပြပါ

   \((-\infty ,\infty )\)

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

   အဖြေကို ဖော်ပြပါ

   When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for \(x.\)

   \[\begin{array}{l}2-x=0 \\ -x=-2 \\ x=2\end{array}\]

   Now, we will exclude 2 from the domain. The answers are all real numbers where \(x<2\) or \(x>2.\) We can use a symbol known as the union, \(\cup ,\) to combine the two sets. In interval notation, we write the solution: \((-\infty ,2)\cup (2,\infty ).\)

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

6. Find the domain of the function: \(f(x)=\frac{1+4x}{2x-1}.\)

   အဖြေကို ဖော်ပြပါ

   \((-\infty ,\frac{1}{2})\cup (\frac{1}{2},\infty )\)

7. Find the domain of the function \(f(x)=\sqrt{7-x}.\)

   အဖြေကို ဖော်ပြပါ

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

   Set the radicand greater than or equal to zero and solve for \(x.\)

   \[\begin{array}{l}7-x\ge 0 \\ -x\ge -7 \\ x\le 7\end{array}\]

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

8. Find the domain of the function \(f(x)=\sqrt{5+2x}.\)

   အဖြေကို ဖော်ပြပါ

   \([-\frac{5}{2},\infty )\)

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

   အဖြေကို ဖော်ပြပါ

   To describe the values, \(x,\) included in the intervals shown, we would say, “ \(x\) is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.”

   Inequality	\(1\le x\le 3\ \text{or}\ x>5\)
   Set-builder notation	\(\{x|1\le x\le 3\ \text{or}\ x>5\}\)
   Interval notation	\([1,3]\cup (5,\infty )\)

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

10. Given , specify the graphed set in

    1. ⓐ words
    2. ⓑ set-builder notation
    3. ⓒ interval notation
    အဖြေကို ဖော်ပြပါ

    1. ⓐ values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3;
    2. ⓑ \(\{x|x\le -2\ \text{or}\ -1\le x<3\}\) ;
    3. ⓒ \((-\infty ,-2]\cup [-1,3)\)

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

    အဖြေကို ဖော်ပြပါ

    We can observe that the horizontal extent of the graph is –3 to 1, so the domain of \(f\) is \((-3,1].\)

    The vertical extent of the graph is 0 to –4, so the range is \([-4,0].\) See .

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

    အဖြေကို ဖော်ပြပါ

    The input quantity along the horizontal axis is “years,” which we represent with the variable \(t\) for time. The output quantity is “thousands of barrels of oil per day,” which we represent with the variable \(b\) for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as \(1973\le t\le 2008\) and the range as approximately \(180\le b\le 2010.\)

    In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

13. Given , identify the domain and range using interval notation.

    အဖြေကို ဖော်ပြပါ

    domain =[1950,2002] range = [47,000,000,89,000,000]

14. Find the domain and range of \(f(x)=2{x}^{3}-x.\)

    အဖြေကို ဖော်ပြပါ

    There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

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

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

    အဖြေကို ဖော်ပြပါ

    We cannot evaluate the function at \(-1\) because division by zero is undefined. The domain is \((-\infty ,-1)\cup (-1,\infty ).\) Because the function is never zero, we exclude 0 from the range. The range is \((-\infty ,0)\cup (0,\infty ).\)

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

    အဖြေကို ဖော်ပြပါ

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

    \[x+4\ge 0\text{ when }x\ge -4\]

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

    We then find the range. We know that \(f(-4)=0,\) and the function value increases as \(x\) increases without any upper limit. We conclude that the range of \(f\) is \([0,\infty ).\)

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

    အဖြေကို ဖော်ပြပါ

    domain: \((-\infty ,2];\) range: \((-\infty ,0]\)

18. A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, \(n,\) to the cost, \(C.\) Since one cannot have fractions of a person, this is really a discrete function. However, for this exercise we will treat it as a continuous function.

    အဖြေကို ဖော်ပြပါ

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

    \[C(n)=\{\begin{array}{lll}5n & \text{if} & 0

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

    \[C(g)=\{\begin{array}{lll}25 & \text{if} & 0Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

    အဖြေကို ဖော်ပြပါ

    To find the cost of using 1.5 gigabytes of data, \(C(1.5),\) we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.

    \[C(1.5)=\text{\$}25\]

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

    \[C(4)=25+10(4-2)=\text{\$}45\]

20. Sketch a graph of the function.

    \[f(x)=\{\begin{array}{lll}{x}^{2} & \text{if} & x\le 1 \\ 3 & \text{if} & 12\end{array}\]
    အဖြေကို ဖော်ပြပါ

    Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality.

    shows the three components of the piecewise function graphed on separate coordinate systems.

    Now that we have sketched each piece individually, we combine them in the same coordinate plane. See .

21. Graph the following piecewise function.

    \[f(x)=\{\begin{array}{lll}{x}^{3} & \text{if} & x<-1 \\ -2 & \text{if} & -14\end{array}\]

22. Why does the domain differ for different functions?

    အဖြေကို ဖော်ပြပါ

    The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

23. How do we determine the domain of a function defined by an equation?

24. Explain why the domain of \(f(x)=\sqrt[3]{x}\) is different from the domain of \(f(x)=\sqrt[]{x}.\)

    အဖြေကို ဖော်ပြပါ

    There is no restriction on \(x\) for \(f(x)=\sqrt[3]{x}\) because you can take the cube root of any real number. So the domain is all real numbers, \((-\infty ,\infty ).\) When dealing with the set of real numbers, you cannot take the square root of negative numbers. So \(x\)-values are restricted for \(f(x)=\sqrt[]{x}\) to nonnegative numbers and the domain is \([0,\infty ).\)

25. When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?

26. How do you graph a piecewise function?

    အဖြေကို ဖော်ပြပါ

    Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the \(x\)-axis and \(y\)-axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate \(-\infty\) or \(\infty .\) Combine the graphs to find the graph of the piecewise function.

27. \(f(x)=-2x(x-1)(x-2)\)

28. \(f(x)=5-2{x}^{2}\)

    အဖြေကို ဖော်ပြပါ

    \((-\infty ,\infty )\)

29. \(f(x)=3\sqrt{x-2}\)

30. \(f(x)=3-\sqrt{6-2x}\)

    အဖြေကို ဖော်ပြပါ

    \((-\infty ,3]\)

31. \(f(x)=\sqrt{4-3x}\)

32. \(\begin{array}{l} \\ f(x)=\sqrt[]{{x}^{2}+4}\end{array}\)

    အဖြေကို ဖော်ပြပါ

    \((-\infty ,\infty )\)

33. \(f(x)=\sqrt[3]{1-2x}\)

34. \(f(x)=\sqrt[3]{x-1}\)

    အဖြေကို ဖော်ပြပါ

    \((-\infty ,\infty )\)

35. \(f(x)=\frac{9}{x-6}\)

36. \(f(x)=\frac{3x+1}{4x+2}\)

    အဖြေကို ဖော်ပြပါ

    \((-\infty ,-\frac{1}{2})\cup (-\frac{1}{2},\infty )\)

37. \(f(x)=\frac{\sqrt{x+4}}{x-4}\)

38. \(f(x)=\frac{x-3}{{x}^{2}+9x-22}\)

    အဖြေကို ဖော်ပြပါ

    \((-\infty ,-11)\cup (-11,2)\cup (2,\infty )\)

39. \(f(x)=\frac{1}{{x}^{2}-x-6}\)

40. \(f(x)=\frac{2{x}^{3}-250}{{x}^{2}-2x-15}\)

    အဖြေကို ဖော်ပြပါ

    \((-\infty ,-3)\cup (-3,5)\cup (5,\infty )\)

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

\binom{n}{k}

binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

e

Euler's number
2.71828…, the base whose exponential is its own derivative.

i

imaginary unit
i² = −1.

\theta

theta
The usual name for an angle.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbf{v},\ \vec{v}

vector
A quantity with magnitude and direction; a column of numbers.

How to: Domain and Range

1. Find the domain of a function defined by an equation.
2. Graph piecewise-defined functions.
3. The smallest term from the interval is written first.
4. The largest term in the interval is written second, following a comma.
5. Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.
6. Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.
7. Identify the input values.
8. Identify any restrictions on the input and exclude those values from the domain.

Questions people ask

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

သင့်ရဲ့ကိုယ်ပိုင်စမ်းသပ်

⌨ Type ✎ Draw

သင်၏သင်္ချာပြဿနာ

ဖြေရှင်းပါ

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from OpenStax Precalculus 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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