Functions and Relations

Find 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 Figure 6.

Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range


Figure 6

We can observe that the graph extends horizontally from \displaystyle -5 to the right without bound, so the domain is \displaystyle \left[-5,\infty \right). The vertical extent of the graph is all range values \displaystyle 5 and below, so the range is \displaystyle \left(\mathrm{-\infty },5\right]. 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.


Find the domain and range of the function \displaystyle f whose graph is shown in Figure 7.

Graph of a function from (-3, 1].


Figure 7


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


This graph has three lines, two straight and one that is curvy. It shows that the domain starts at -3 (not including -3) all the way to 1. The range shows from -4 to 0.


Figure 8


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

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