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## Subsection1.3Functions Defined by Graphs

A graph may also be used to define one variable as a function of another. The input variable is displayed on the horizontal axis, and the output variable on the vertical axis.

###### Example1.5

Figure1.6 shows the number of hours, $H\text{,}$ that the sun is above the horizon in Peoria, Illinois, on day $t\text{,}$ where January 1 corresponds to $t = 0\text{.}$

1. The input variable, $t\text{,}$ appears on the horizontal axis. The number of daylight hours, $H\text{,}$ is a function of the date. The output variable appears on the vertical axis.

2. The point on the curve where $t = 150$ has $H \approx 14.1\text{,}$ so Peoria gets about 14.1 hours of daylight when $t = 150\text{,}$ which is at the end of May.

3. $H = 12$ at the two points where $t \approx 85$ (in late March) and $t \approx 270$ (late September).

4. The maximum value of 14.4 hours occurs on the longest day of the year, when $t \approx 170\text{,}$ about three weeks into June. The minimum of 9.6 hours occurs on the shortest day, when $t \approx 355\text{,}$ about three weeks into December. We have a method of quickly determining if a relationship is a function once we have a graph of the relationship.

###### The Vertical Line Test

A graph represents a function if and only if every vertical line intersects the graph in at most one point.

###### Example1.8

Use the vertical line test to decide which of the graphs in Figure1.9 represent functions.

• Graph (b) is not the graph of a function, because the vertical line at (for example) $x = 1$ intersects the graph at two points.
• For graph (c), notice the break in the curve at $x = 2\text{:}$ The solid dot at $(2, 1)$ is the only point on the graph with $x = 2\text{;}$ the open circle at $(2, 3)$ indicates that $(2, 3)$ is not a point on the graph. Thus, graph (c) is a function, with $f(2) = 1\text{.}$