###### Example140

Graph the function defined by

Think of the plane as divided into two regions by the vertical line \(x = 1\text{,}\) as shown in Figure141. In the left-hand region (\(x \le 1\)), we graph the line \(y = x + 1\text{.}\) (The fastest way to graph the line is to plot its intercepts, \((-1, 0)\) and \((0, 1)\text{.}\))

Notice that the value \(x = 1\) is included in the first region, so \(f (1) = 1 + 1 = 2\text{,}\) and the point \((1, 2)\) is included on the graph. We indicate this with a solid dot at the point \((1, 2)\text{.}\)

In the right-hand region (\(x \gt 1\)), we graph the horizontal line \(y = 3\text{.}\) The value \(x = 1\) is not included in the second region, so the point \((1, 3)\) is not part of the graph. We indicate this with an open circle at the point \((1, 3)\text{.}\)