Question

Find the functions (a) \({\bf{f}} \circ {\bf{g}}\) ,(b) \({\bf{g}} \circ {\bf{f}}\) ,(c) \({\bf{f}} \circ {\bf{f}}\) and (d) \({\bf{g}} \circ {\bf{g}}\) and their domains.

\({\bf{f}}\left( {\bf{x}} \right){\bf{ = x - 2}}\) \({\bf{g}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{2}}}{\bf{ + 3x + 4}}\)

Short answer

(a) The composition \({\bf{f}} \circ {\bf{g}}\) with \({\bf{f}}\left( {\bf{x}} \right){\bf{ = x - 2}}\) and \({\bf{g}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{2}}}{\bf{ + 3x + 4}}\) is \({{\bf{x}}^{\bf{2}}}{\bf{ + 3x + 2}}\) and has domain\(\left( { - \infty ,\infty } \right)\).

(b) The composition \({\bf{g}} \circ {\bf{f}}\) with \({\bf{f}}\left( {\bf{x}} \right){\bf{ = x - 2}}\) and \({\bf{g}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{2}}}{\bf{ + 3x + 4}}\) is \({{\bf{x}}^{\bf{2}}}{\bf{ - x + 2}}\) and has domain\(\left( { - \infty ,\infty } \right)\).

(c) The composition \({\bf{f}} \circ {\bf{f}}\) with \({\bf{f}}\left( {\bf{x}} \right){\bf{ = x - 2}}\) and \({\bf{g}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{2}}}{\bf{ + 3x + 4}}\) is \({\bf{x - 4}}\) and has domain\(\left( { - \infty ,\infty } \right)\).

(d) The composition \({\bf{g}} \circ {\bf{g}}\) with \({\bf{f}}\left( {\bf{x}} \right){\bf{ = x - 2}}\) and \({\bf{g}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{2}}}{\bf{ + 3x + 4}}\) is \({{\bf{x}}^{\bf{4}}}{\bf{ + 6}}{{\bf{x}}^{\bf{3}}}{\bf{ + 20}}{{\bf{x}}^{\bf{2}}}{\bf{ + 33x + 32}}\) and has domain\(\left( { - \infty ,\infty } \right)\).

Step-by-step solution

Given data

The given functions are;

\(f\left( x \right) = x - 2\)and \(g\left( x \right) = {x^2} + 3x + 4\)

Composite Function

A composite function is formed when one function is substituted into another function.For two functions\(f\left( x \right)\)and\(g\left( x \right)\), their composition is defined as;

\({\bf{fog = f}}\left( {{\bf{g}}\left( {\bf{x}} \right)} \right)\)

Determine the composition function \({\bf{f}} \circ {\bf{g}}\) 

Evaluate\({\bf{f}} \circ {\bf{g}}\)

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

This function is well defined everywhere. Thus, the domain is\(\left( { - \infty ,\infty } \right)\).

Hence, the composition \(f \circ g\) is \({x^2} + 3x + 2\) with the domain\(\left( { - \infty ,\infty } \right)\).

Determine the composition function\({\bf{g}} \circ {\bf{f}}\) 

Evaluate\({\bf{g}} \circ {\bf{f}}\)

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

This function is well defined everywhere. Thus, the domain is\(\left( { - \infty ,\infty } \right)\).

Hence, the composition \(g \circ f\) is \({x^2} - x + 2\) with the domain\(\left( { - \infty ,\infty } \right)\).

Determine the composition function\({\bf{f}} \circ {\bf{f}}\) 

Evaluate\({\rm{f}} \circ {\bf{f}}\)

\(\begin{aligned}f \circ f &= f\left( {f\left( x \right)} \right)\\ &= f\left( {x - 2} \right)\\ &= \left( {x - 2} \right) - 2\\ &= x - 4\end{aligned}\)

This function is well defined everywhere. Thus, the domain is\(\left( { - \infty ,\infty } \right)\).

Hence, the composition \(f \circ f\) is \(x - 4\) with the domain\(\left( { - \infty ,\infty } \right)\).

Determine the composition function \({\bf{g}} \circ {\bf{g}}\) 

Now evaluate\({\rm{g}} \circ {\rm{g}}\)

\(\begin{aligned}g \circ g &= g\left( {g\left( x \right)} \right)\\ &= g\left( {{x^2} + 3x + 4} \right)\\ &= {\left( {{x^2} + 3x + 4} \right)^2} + 3\left( {{x^2} + 3x + 4} \right) + 4\\ &= {x^4} + 9{x^2} + 16 + 6{x^3} + 8{x^2} + 24x + 3{x^2} + 9x + 12 + 4\\ &= {x^4} + 6{x^3} + 20{x^2} + 33x + 32\end{aligned}\)

This function is well defined everywhere. Thus, the domain is\(\left( { - \infty ,\infty } \right)\).

Hence, the composition \(g \circ g\) is \({x^4} + 6{x^3} + 20{x^2} + 33x + 32\) with the domain\(\left( { - \infty ,\infty } \right)\).