Question

Find \(f \circ g \circ h\).

42. \(f\left( x \right) = \left| {x - {\bf{4}}} \right|\), \(g\left( x \right) = {{\bf{2}}^x}\), \(h\left( x \right) = \sqrt x \)

Short answer

The function is \(\left| {{2^{\sqrt x }} - 4} \right|\).

Step-by-step solution

Find the function \(g \circ h\)

The function \(g \circ h\left( x \right)\) can be evaluated as follows:

\(\begin{aligned}g \circ h\left( x \right) &= g\left[ {h\left( x \right)} \right]\\ &= g\left[ {\sqrt x } \right]\\ &= {2^{\sqrt x }}\end{aligned}\)

Thus, the function is \(g \circ h\left( x \right) = {2^{\sqrt x }}\).

Find the function \(f \circ g \circ h\)

The function \(f \circ g \circ h\) can be evaluated as follows:

\(\begin{aligned}f \circ g \circ h\left( x \right) &= f\left[ {{2^{\sqrt x }}} \right]\\ &= \left| {{2^{\sqrt x }} - 4} \right|\end{aligned}\)

So, thecomposite functionis \(\left| {{2^{\sqrt x }} - 4} \right|\).