Question

Express the function in the form \({\bf{f}} \circ {\bf{g}}\)

\({\bf{F}}\left( {\bf{x}} \right){\bf{ = }}{\left( {{\bf{2x + }}{{\bf{x}}^{\bf{2}}}} \right)^{\bf{4}}}\)

Short answer

The function \({\bf{F}}\left( {\bf{x}} \right){\bf{ = }}{\left( {{\bf{2x + }}{{\bf{x}}^{\bf{2}}}} \right)^{\bf{4}}}\)can be expressed in the form \({\bf{f}} \circ {\bf{g}}\)with\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{4}}}\) and \({\bf{g}}\left( {\bf{x}} \right){\bf{ = 2x + }}{{\bf{x}}^{\bf{2}}}\).

Step-by-step solution

Given data

The provided function is

\(F\left( x \right) = {\left( {2x + {x^2}} \right)^4}\)

Step 2:

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

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

The composition \({\bf{f}} \circ {\bf{g}} = {\bf{F}}\) 

Let \(f\left( x \right) = {x^4}\) and \(g\left( x \right) = 2x + {x^2}\)

From equation (1),

\(\begin{aligned}f \circ g &= f\left( {2x + {x^2}} \right)\\ &= {\left( {2x + {x^2}} \right)^4}\\ &= F\left( x \right)\end{aligned}\)

Hence, the required functions are \(f\left( x \right) = {x^4}\)and \(g\left( x \right) = 2x + {x^2}\).