Find domain and range from a graph, and an equation.

Requite the domain and range of the toolkit functions.

Another manner to place the domain and range of functions is by using graphs. Considering the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the [latex]x[/latex]-axis. The range is the set of possible output values, which are shown on the [latex]y[/latex]-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.

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

Example: Finding Domain and Range from a Graph

Find the domain and range of the part [latex]f[/latex].

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

Find the domain and range of the function [latex]f[/latex].

Attempt It

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

Prove Solution

Domain = [latex][1950, 2002][/latex] Range = [latex][47,000,000, 89,000,000][/latex]

Yes. For example, the domain and range of the cube root office are both the set of all existent numbers.

Domain and Range of Toolkit Functions

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

For the constant function
[latex]f\left(ten\right)=c[/latex], the domain consists of all existent numbers; in that location are no restrictions on the input. The but output value is the abiding [latex]c[/latex], and so the range is the prepare [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].

For the identity part
[latex]f\left(ten\right)=10[/latex], in that location is no brake on [latex]ten[/latex]. Both the domain and range are the set of all real numbers.

For the absolute value function
[latex]f\left(10\correct)=|10|[/latex], there is no brake on [latex]x[/latex]. Notwithstanding, because absolute value is divers as a distance from 0, the output can only be greater than or equal to 0.

For the quadratic function
[latex]f\left(x\correct)={10}^{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.

For the cubic function
[latex]f\left(x\right)={ten}^{3}[/latex], the domain is all real numbers considering the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, and then the domain and range include all real numbers.

For the reciprocal office
[latex]f\left(ten\right)=\frac{1}{x}[/latex], we cannot divide by 0, so nosotros must exclude 0 from the domain. Further, i divided by any value can never be 0, so the range also volition not include 0. In set-builder notation, we could likewise write [latex]\left\{x|\text{ }x\ne 0\right\}[/latex], the set of all real numbers that are not nothing.

For the reciprocal squared role
[latex]f\left(10\correct)=\frac{1}{{ten}^{2}}[/latex], we cannot divide by [latex]0[/latex], then we must exclude [latex]0[/latex] from the domain. There is also no [latex]x[/latex] that can requite 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, then the range includes only positive numbers.

For the foursquare root part
[latex]f\left(x\correct)=\sqrt[]{x}[/latex], we cannot take the foursquare root of a negative real number, then 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 be positive, even though the square of the negative number [latex]-\sqrt{10}[/latex] also gives us [latex]10[/latex].

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

Try It

Contribute!

Did you have an thought for improving this content? We’d love your input.