GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 61

Let $S$ be the set of all functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Consider the two binary operations + and $\circ$ on $S$ defined as pointwise addition and composition of functions, as follows.
$$
\begin{gathered}
(f+g)(x)=f(x)+g(x) \\
(f \circ g)(x)=f(g(x))
\end{gathered}
$$
Which of the following statements are true?

1. $\circ$ is commutative.
2. + and $\circ$ satisfy the left distributive law $f \circ(g+h)=(f \circ g)+(f \circ h)$.
3. + and $\circ$ satisfy the right distributive law $(g+h) \circ f=(g \circ f)+(h \circ f)$.

1. None
2. III only
3. II and III only
4. I, II, and III