Question

Find\(fogoh\)

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

Short answer

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

Step-by-step solution

Given data

The provided functions are:

\(f\left( x \right) = \sqrt {x - 3} \)

\(g\left( x \right) = {x^2}\)

\(h\left( x \right) = {x^3} + 2\)

definition of composition

For two functions \(f\left( x \right)\), \(g\left( x \right)\)and \(h\left( x \right)\) their composition is defined as

\({\bf{f}} \circ {\bf{g}} \circ {\bf{h = f}}\left( {{\bf{g}}\left( {{\bf{h}}\left( {\bf{x}} \right)} \right)} \right)\;\;\;\;\;.....\left( {\bf{1}} \right)\)

The composition \(fogoh\)

From equation (1),

\(\begin{aligned}{c}f \circ g \circ h &= f\left( {g\left( {{x^3} + 2} \right)} \right)\\ &= f\left( {{{\left( {{x^3} + 2} \right)}^2}} \right)\\ &= \sqrt {{{\left( {{x^3} + 2} \right)}^2} - 3} \\ &= \sqrt {{x^6} + 4{x^3} + 1} \end{aligned}\)

Hence, the composition \(f \circ g \circ h\) is \(\sqrt {{x^6} + 4{x^3} + 1} \).