Question

let $f(x) = 3x + 5$ and $g(x) = 5x^2 + 5x$. after simplifying, $(f \circ g)(x) = \square$ $(g \circ f)(x) = \square$ question help: video written example

Explanation:

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

Step1: Recall composition definition

The composition \((f \circ g)(x)\) means \(f(g(x))\). So we substitute \(g(x)\) into \(f(x)\).
Given \(f(x) = 3x + 5\) and \(g(x)=5x^{2}+5x\), we replace \(x\) in \(f(x)\) with \(g(x)\).

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

\(f(g(x))=3(g(x)) + 5\). Now substitute \(g(x)=5x^{2}+5x\) into this:
\(f(g(x)) = 3(5x^{2}+5x)+5\)

Step3: Distribute and simplify

First, distribute the 3: \(3\times5x^{2}+3\times5x + 5=15x^{2}+15x + 5\)

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

Step1: Recall composition definition

The composition \((g \circ f)(x)\) means \(g(f(x))\). So we substitute \(f(x)\) into \(g(x)\).
Given \(g(x)=5x^{2}+5x\) and \(f(x) = 3x + 5\), we replace \(x\) in \(g(x)\) with \(f(x)\).

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

\(g(f(x))=5(f(x))^{2}+5(f(x))\). Now substitute \(f(x)=3x + 5\) into this:
\(g(f(x))=5(3x + 5)^{2}+5(3x + 5)\)

Step3: Expand and simplify

First, expand \((3x + 5)^{2}\) using the formula \((a + b)^{2}=a^{2}+2ab + b^{2}\), where \(a = 3x\) and \(b = 5\). So \((3x + 5)^{2}=(3x)^{2}+2\times3x\times5+5^{2}=9x^{2}+30x + 25\)
Then, \(5(9x^{2}+30x + 25)+5(3x + 5)=45x^{2}+150x+125 + 15x + 25\)
Combine like terms: \(45x^{2}+(150x + 15x)+(125 + 25)=45x^{2}+165x + 150\)

Answer:

\((f \circ g)(x)=\boldsymbol{15x^{2}+15x + 5}\)
\((g \circ f)(x)=\boldsymbol{45x^{2}+165x + 150}\)