# Determine Domain and Range from a Graph

### Learning Outcomes

- Find domain and range from a graph, and an equation.
- Give the domain and range of the toolkit functions.

### Example: Finding Domain and Range from a Graph

Find the domain and range of the function [latex]f[/latex].Answer: We can observe that the horizontal extent of the graph is –3 to 1, so the domain of [latex]f[/latex] is [latex]\left(-3,1\right][/latex]. The vertical extent of the graph is 0 to [latex]–4[/latex], so the range is [latex]\left[-4,0\right][/latex].

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

Find the domain and range of the function [latex]f[/latex].Answer: The input quantity along the horizontal axis is "years," which we represent with the variable [latex]t[/latex] for time. 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 right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [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 [latex][1973, 2008][/latex], and the range is about [latex][180, 2010][/latex]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

### Try It

Given the graph, identify the domain and range using interval notation.Answer: Domain = [latex][1950, 2002][/latex] Range = [latex][47,000,000, 89,000,000][/latex]

[ohm_question]3765[/ohm_question]### Q & A

**Can a function’s domain and range be the same?**

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

## Domain and Range of Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.**constant function**[latex]f\left(x\right)=c[/latex], the domain consists of all real numbers; there are no restrictions on the input. The only 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].

**identity function**[latex]f\left(x\right)=x[/latex], there is no restriction on [latex]x[/latex]. Both the domain and range are the set of all real numbers.

**absolute value function**[latex]f\left(x\right)=|x|[/latex], there is no restriction on [latex]x[/latex]. However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.

**quadratic function**[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 any negative values for the range, the range is only nonnegative real numbers.

**cubic function**[latex]f\left(x\right)={x}^{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 real numbers.

**reciprocal function**[latex]f\left(x\right)=\frac{1}{x}[/latex], we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write [latex]\left\{x|\text{ }x\ne 0\right\}[/latex], the set of all real numbers that are not zero.

**reciprocal squared function**[latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex], we cannot divide by [latex]0[/latex], so we must exclude [latex]0[/latex] from the domain. There is also no [latex]x[/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.

**square root function**[latex]f\left(x\right)=\sqrt[]{x}[/latex], we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number [latex]x[/latex] is defined to be positive, even though the square of the negative number [latex]-\sqrt{x}[/latex] also gives us [latex]x[/latex].

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

### tip for success

When evaluating functions to plot points, remember to wrap the variable in parentheses to be sure you handle negative input appropriately. Ex. For the function [latex]f(x) = x^2[/latex], evaluate [latex]f(-3)[/latex].Answer: [latex]\begin{align}f(-3) &= (-3)^2 \\ &= 9\end{align}[/latex].

### Try It

[ohm_question]47481[/ohm_question] [ohm_question]47483[/ohm_question] [ohm_question]47484[/ohm_question] [ohm_question]47487[/ohm_question]## Licenses & Attributions

### CC licensed content, Original

- Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution. - Question ID 47487.
**Authored by:**Lumen Learning.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC-BY + GPL.

### CC licensed content, Shared previously

- College Algebra.
**Provided by:**OpenStax**Authored by:**Abramson, Jay et al..**Located at:**https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites.**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected]. - Question ID 3765.
**Authored by:**Jenck,Michael.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC-BY + GPL. - Question ID 47481, 47483, 47484.
**Authored by:**Day, Alyson.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC-BY + GPL.