GATE CS 2023 Q9 — General Aptitude

Given that $f(x) = g(y)$ for all values of $x$ and $y$, where $f(x)$ is a function of $x$ alone and $g(y)$ is a function of $y$ alone.

Let us analyze the behavior of these functions by keeping one variable constant:

1. Fix $y$: Let $y$ be fixed at some arbitrary value $y_0$. Since $g(y)$ is a function of $y$, $g(y_0)$ is a constant value, say $C$.
   Since the relation $f(x) = g(y)$ must hold for all $x$ and $y$, substituting $y = y_0$ gives: $$f(x) = g(y_0) = C \quad \text{for all } x$$ This means $f(x)$ must be a constant function.
2. Fix $x$: Similarly, let $x$ be fixed at some arbitrary value $x_0$. Then $f(x_0)$ is a constant.
   Substituting $x = x_0$ into the relation gives: $$g(y) = f(x_0) = C \quad \text{for all } y$$ This means $g(y)$ must also be a constant function.

Therefore, both $f(x)$ and $g(y)$ must be equal to the same constant for all $x$ and $y$:

Hence, option B is necessarily true.