Question

if $f(x) = 5x + 2$ and $g(x) = x + 3$, find $(f \circ g)(x)$ and $(g \circ f)(x)$.

$(f \circ g)(x) = \square$

Explanation:

Step1: Recall function composition

Function composition \((f \circ g)(x)\) means \(f(g(x))\). So we substitute \(g(x)\) into \(f(x)\).

Step2: Substitute \(g(x)\) into \(f(x)\)

Given \(f(x) = 5x + 2\) and \(g(x)=x + 3\), replace \(x\) in \(f(x)\) with \(g(x)=x + 3\). So \(f(g(x))=5(g(x))+2\).

Step3: Replace \(g(x)\) with its expression

Substitute \(g(x)=x + 3\) into the above, we get \(f(g(x)) = 5(x + 3)+2\).

Step4: Simplify the expression

Expand \(5(x + 3)\) to get \(5x+15\), then add 2: \(5x + 15+2=5x+17\).

Answer:

\(5x + 17\)