We are given that $g \circ f$ (the composition of $g$ and $f$) is surjective, and we need to determine whether $f$ or $g$ must also be surjective, justifying the answers.

Recall Definitions:

• A function is surjective (onto) if every element of the codomain has a preimage in the domain. In other words, for a function $h: A \to B$, for every $b \in B$, there exists some $a \in A$ such that $h(a) = b$.

Now, let’s examine each case.

1. Is $f$ surjective?

To answer this, we need to consider the relationship between $g \circ f$ being surjective and the properties of $f$.

• Let $f: A \to B$ and $g: B \to C$.
• Since $g \circ f$ is surjective, for every $c \in C$, there exists some $a \in A$ such that $g(f(a)) = c$.

However, this does not imply that $f$ itself is surjective. The reason is that $f$ maps from $A$ to $B$, but the fact that $g \circ f$ is surjective tells us only that the elements of $C$ are covered, not that all elements of $B$ are covered by $f$.

Thus, $f$ is not necessarily surjective.

2. Is $g$ surjective?

Yes, $g$ must be surjective.

Here's why: Since $g \circ f$ is surjective, for every $c \in C$, there exists some $a \in A$ such that $g(f(a)) = c$. This implies that for every $c \in C$, there is some element $b = f(a) \in B$ such that $g(b) = c$. Thus, $g$ must cover all elements of $C$ by taking preimages from $B$, which means $g$ is surjective.

Summary:

• $f$ is not necessarily surjective.
• $g$ must be surjective.

Do you want further details or have any questions?

Here are 5 related questions to expand on this concept:

1. What would happen if $f$ were injective but not surjective in this scenario?
2. Can you give an example where $g \circ f$ is surjective but $f$ is not?
3. How does injectivity of $g$ affect the surjectivity of $g \circ f$?
4. What if both $f$ and $g$ were injective—how would that impact $g \circ f$?
5. How do surjective and injective functions differ in terms of their composition?

Tip: When working with compositions of functions, always carefully consider how the properties (injectivity, surjectivity) of each function impact the overall behavior of the composition.