Question

Use a graphing utility to graph the function. Use the graph to determine any \(x\) -values at which the function is not continuous. $$ h(x)=\frac{1}{x^{2}-x-2} $$

Short answer

The function \(h(x) = \frac{1}{x^{2}-x-2}\) is not continuous at \(x = 2\) and \(x = -1\).

Step-by-step solution

Identify the Critical Points

Set the denominator of the function equal to zero and solve for \(x\). For this function, \(x^{2}-x-2 = 0\). Use the quadratic formula to solve for \(x\), which gives \(x = 2\) and \(x = -1\).

Graph the Function

Use a graphing utility to graph the function \(h(x)=\frac{1}{x^{2}-x-2}\). It can be observed that there will be vertical asymptotes at the critical points, \(x = 2\) and \(x = -1\), since the value of \(h(x)\) approaches infinity as \(x\) approaches these values.

Identify the Points of Discontinuity

From the graph, it can be observed that vertical asymptotes occur at \(x = 2\) and \(x = -1\). Thus, these are the x-values at which the function is not continuous.