Question

let $f(x) = x - 2$ and $g(x) = x^2 + 3x + 4$. calculate $(g circ f)(x)$
$\bigcirc$ $x^2 + 3x + 2$
$\bigcirc$ $x^2 - x + 2$
$\bigcirc$ $x^2 + 3x - 2$
$\bigcirc$ $x^2 + 7x + 2$
$\bigcirc$ $x^2 - x - 2$

Explanation:

Step1: Recall composition of functions

The composition \((g \circ f)(x)\) means \(g(f(x))\). So we need to substitute \(f(x)\) into \(g(x)\).
Given \(f(x)=x - 2\) and \(g(x)=x^{2}+3x + 4\), we substitute \(x\) in \(g(x)\) with \(f(x)=x - 2\).

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

\[

$$\begin{align*} g(f(x))&=(x - 2)^{2}+3(x - 2)+4\\ &=x^{2}-4x + 4+3x-6 + 4\\ &=x^{2}-x + 2 \end{align*}$$

\]

Answer:

\(x^{2}-x + 2\) (corresponding to the option: \(x^{2}-x + 2\))