Question

question 14
if ( f(x) = x^2 + 4 ), ( g(x) = x - 5 ), and ( h(x) = sqrt{x} ) then
( (f circ g)(x) = )
( (g circ f)(x) = )
( (h circ g)(x) = )

Explanation:

For \((f \circ g)(x)\):

Step 1: Recall the definition of function composition

Function composition \((f \circ g)(x)\) means \(f(g(x))\). So we need to substitute \(g(x)\) into \(f(x)\).
Given \(f(x)=x^{2}+4\) and \(g(x)=x - 5\), we substitute \(g(x)\) (which is \(x - 5\)) for every \(x\) in \(f(x)\).

Step 2: Substitute and simplify

\(f(g(x))=f(x - 5)=(x - 5)^{2}+4\)
Expand \((x - 5)^{2}\) using the formula \((a - b)^{2}=a^{2}-2ab + b^{2}\), where \(a = x\) and \(b = 5\).
\((x - 5)^{2}=x^{2}-10x + 25\)
So \(f(g(x))=x^{2}-10x + 25+4=x^{2}-10x + 29\)

For \((g \circ f)(x)\):

Step 1: Recall the definition of function composition

Function composition \((g \circ f)(x)\) means \(g(f(x))\). So we need to substitute \(f(x)\) into \(g(x)\).
Given \(g(x)=x - 5\) and \(f(x)=x^{2}+4\), we substitute \(f(x)\) (which is \(x^{2}+4\)) for every \(x\) in \(g(x)\).

Step 2: Substitute and simplify

\(g(f(x))=g(x^{2}+4)=(x^{2}+4)-5\)
Simplify the expression: \(x^{2}+4 - 5=x^{2}-1\)

For \((h \circ g)(x)\):

Step 1: Recall the definition of function composition

Function composition \((h \circ g)(x)\) means \(h(g(x))\). So we need to substitute \(g(x)\) into \(h(x)\).
Given \(h(x)=\sqrt{x}\) and \(g(x)=x - 5\), we substitute \(g(x)\) (which is \(x - 5\)) for every \(x\) in \(h(x)\).

Step 2: Substitute and simplify

\(h(g(x))=h(x - 5)=\sqrt{x - 5}\) (we assume \(x-5\geq0\) i.e., \(x\geq5\) for the square root to be defined in real numbers)

Answer:

s:
\((f \circ g)(x)=\boldsymbol{x^{2}-10x + 29}\)
\((g \circ f)(x)=\boldsymbol{x^{2}-1}\)
\((h \circ g)(x)=\boldsymbol{\sqrt{x - 5}}\) (for \(x\geq5\))