Consider functions $f$ and $g$ such that the composite function $g \circ f$ is defined and is one-one. Are $f$ and $g$ both necessarily one-one?

(NO) To determine if $f$ and $g$ are necessarily one-one,let us analyze the properties of the composite function $g \circ f$.
If $g \circ f$ is one-one,then for any $x_1, x_2$ in the domain of $f$,$g(f(x_1)) = g(f(x_2))$ implies $f(x_1) = f(x_2)$,which further implies $x_1 = x_2$. This confirms that $f$ must be one-one.
However,$g$ does not necessarily have to be one-one.
Consider the functions $f: \{1, 2, 3, 4\} \rightarrow \{1, 2, 3, 4, 5, 6\}$ defined as $f(x) = x$ for all $x$,and $g: \{1, 2, 3, 4, 5, 6\} \rightarrow \{1, 2, 3, 4, 5, 6\}$ defined as $g(x) = x$ for $x \in \{1, 2, 3, 4\}$ and $g(5) = g(6) = 5$.
Here,$(g \circ f)(x) = x$ for all $x \in \{1, 2, 3, 4\}$,which is clearly one-one.
However,$g$ is not one-one because $g(5) = g(6) = 5$ while $5 \neq 6$.
Therefore,$f$ must be one-one,but $g$ is not necessarily one-one.