Question

given $f(x) = x + 1$ and $g(x) = x^2$, what is $(g \circ f)(x)$?
$(g \circ f)(x) = x^2 + 1$
$(g \circ f)(x) = x^2(x + 1)$
$(g \circ f)(x) = (x + 1)^2$
$(g \circ f)(x) = x^2 + x + 1$

Explanation:

Step1: Recall function composition

Function composition \((g \circ f)(x)\) means \(g(f(x))\). So we first find \(f(x)\) and then substitute it into \(g\).

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

Given \(f(x)=x + 1\) and \(g(x)=x^{2}\), substitute \(f(x)\) into \(g\): \(g(f(x))=g(x + 1)\).
Now, replace \(x\) in \(g(x)\) with \(x + 1\). Since \(g(x)=x^{2}\), then \(g(x + 1)=(x + 1)^{2}\).

Answer:

\((g \circ f)(x)=(x + 1)^{2}\) (the third option)