Question

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

Explanation:

Step1: Recall composition of functions

The composition \( (g \circ f)(x) \) means \( g(f(x)) \). So we first find \( f(x) \) and then substitute it into \( g(x) \).
Given \( f(x) = x + 1 \) and \( g(x)=x^2 \).

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

We substitute \( f(x)=x + 1 \) into \( g(x) \), so wherever there is an \( x \) in \( g(x) \), we replace it with \( f(x)=x + 1 \).
Since \( g(x)=x^2 \), then \( g(f(x))=g(x + 1)=(x + 1)^2 \).

Answer:

\((g \circ f)(x)=(x + 1)^2\) (the first option among the given choices)