GATE CS 2023 Q.9: Quantitative Aptitude

$f(x)$ and $g(y)$ are functions of $x$ and $y$ , respectively, and $f(x) = g(y)$ for all real values of $x$ and $y$ . Which one of the following options is necessarily TRUE for all $x$ and $y$ ?

$f(x) = 0$ and $g(y) = 0$

$f(x) = g(y) = \text{constant}$

$f(x) \neq \text{constant}$ and $g(y) \neq \text{constant}$

$f(x) + g(y) = f(x) - g(y)$

Official answer and solution

Official answer: B
From: Official GATE 2023 answer key

Solution

Use the independence of variables $x$ and $y$ to deduce the nature of $f$ and $g$ .

We are given $f(x) = g(y)$ for all real $x$ and $y$ .

Fix an arbitrary value of $y$ , say $y = 0$ . Let $g(0) = c$ (a constant real number). Then for all real $x$ , $f(x) = g(0) = c$ . Hence $f(x)$ is identically constant.

Similarly, fix an arbitrary value of $x$ , say $x = 0$ . Then for all real $y$ , $g(y) = f(0) = c$ . Hence $g(y)$ is identically constant and equals the same constant $c$ .

Therefore, $f(x) = g(y) = \text{constant}$ is necessarily true for all real $x$ and $y$ .

Final answer

Option (B)

Common trap

Assuming the constant must be zero (Option A), whereas $f(x) = g(y) = 5$ is equally valid.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2023 Computer Science and Information Technology (CS)
Question: 9 of 65
Section: General Aptitude

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