Question

Use the given graphs of \(f\) and \(g\) to estimate the value of \(f\left( {g\left( x \right)} \right)\) for \(x = - {\bf{5}},{\rm{ }} - {\bf{4}},{\rm{ }} - {\bf{3}},...,{\bf{5}}\). Use these estimates to sketch a rough graph of \(f \circ g\).

Short answer

The values of \(f\left( {g\left( x \right)} \right)\) are listed in the table as shown below:

The graph is shown below:

Step-by-step solution

The composite function

Consider two functions \(f\) and \(g\), the composite function\(f \circ g\)(also called the composition of\(f\) and \(g\)) is defined as:

\(\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right)\)

Estimate the value of \(f\left( {g\left( x \right)} \right)\) for \(x =  - 5, - 4, - 3,...,5\)

a)

It is observed from the graph that \(g\left( 0 \right) \approx 2.8\) and \(f\left( {2.8} \right) \approx - 0.5\).

Determine a particular value of \(f\left( {g\left( x \right)} \right)\) for \(x = 0\) is shown below:

\(\begin{aligned}f\left( {g\left( 0 \right)} \right) = f\left( {2.8} \right)\\ \approx - 0.5\end{aligned}\)

The other values were determined in the same way are listed in the table as shown below:

Use the table value to sketch the graph of as shown below:

Use the table value to sketch the graph of as shown below:

Thus, the graph is obtained.