Question

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

41. \(f\left( x \right) = {\bf{3}}x - {\bf{2}}\), \(g\left( x \right) = {\bf{sin}}x\), \(h\left( x \right) = {x^{\bf{2}}}\)

Short answer

The function is \(3\sin \left( {{x^2}} \right) - 2\).

Step-by-step solution

Find the function \(g \circ h\)

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

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

Thus, \(g \circ h\left( x \right) = \sin \left( {{x^2}} \right)\).

Find the function \(f \circ g \circ h\) and its domain

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

\(\begin{aligned}f \circ g \circ h\left( x \right) &= f\left[ {\sin {x^2}} \right]\\ &= 3\left( {\sin \left( {{x^2}} \right)} \right) - 2\\ &= 3\sin \left( {{x^2}} \right) - 2\end{aligned}\)

So, thecomposite functionis \(3\sin \left( {{x^2}} \right) - 2\).