Question

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

Explanation:

Step1: Recall function composition

The composition \( (g \circ f)(x) \) means \( g(f(x)) \).

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

We know \( f(x) = x + 1 \) and \( g(x) = x^2 \). So substitute \( f(x) \) (which is \( x + 1 \)) into \( g(x) \) in place of \( x \). That gives \( g(f(x)) = (f(x))^2 = (x + 1)^2 \).

Answer:

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