Question

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

43. \(f\left( x \right) = \sqrt {x - {\bf{3}}} \), \(g\left( x \right) = {x^{\bf{2}}}\), \(h\left( x \right) = {x^{\bf{3}}} + {\bf{2}}\)

Short answer

The function is \(\sqrt {{x^6} + 4{x^3} + 1} \).

Step-by-step solution

Find the function \(g \circ h\)

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

\(\begin{aligned}g \circ h\left( x \right) &= g\left( {h\left( x \right)} \right)\\ &= g\left( {{x^3} + 2} \right)\\ &= {\left( {{x^3} + 2} \right)^2}\\ &= {x^6} + 4{x^3} + 4\end{aligned}\)

Thus, the function is \(g \circ h\left( x \right) = {x^6} + 4{x^3} + 4\).

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

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

\(\begin{aligned}f \circ g \circ h\left( x \right) &= f\left( {{x^6} + 4{x^3} + 4} \right)\\ &= \sqrt {{x^6} + 4{x^3} + 4 - 3} \\ &= \sqrt {{x^6} + 4{x^3} + 1} \end{aligned}\)

So, the composite function is \(\sqrt {{x^6} + 4{x^3} + 1} \).