Question

Discuss the continuity of the composite function \(h(x)=f(g(x))\). $$ \begin{array}{l} f(x)=\frac{1}{x-6} \\ g(x)=x^{2}+5 \end{array} $$

Short answer

The composite function \(h(x) = f(g(x))\) is discontinuous at \(x = 1\) and \(x = -1\), as these are the points that make the output of \(g(x)\) equal 6, leading to a discontinuity in the composite function due to the discontinuity of \(f(x)\) at \(x = 6\).

Step-by-step solution

Analyze the Discontinuity in f(x)

The function \(f(x) = \frac{1}{x-6}\) is discontinuous at \(x = 6\). This means there exists a point in \(h(x) = f(g(x))\) that could be potentially discontinuous.

Identify the critical x-value in g(x)

Since \(f(x)\) is discontinuous when \(x = 6\), we need to find for which x-value in \(g(x)\), the output is 6. Solving \(g(x) = x^{2} + 5 = 6\), we get \(x = \pm 1\). Both these x-values could correspond to potential points of discontinuity in the composite function \(h(x)\).

Check Continuity of h(x)

Assess the continuity of \(h(x)\) at \(x = 1\) and \(x = -1\) by calculating the limits. If the result for \(x = \pm 1\) is that the limit and function value both exist and are equal, then \(h(x)\) is continuous in these points. If not, then \(h(x)\) is discontinuous in these points.