GATE CS 2023 Set 1 — Question 9
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Question
and are functions of and , respectively, and for all real values of and . Which one of the following options is necessarily TRUE for all and ?
Correct answer
(B) f(x) = g(y) = constant
Solution
Given that for all real values of and .Let's fix at some value . Then for all . Since is a fixed real number, must be a constant function.
Similarly, let's fix at some value . Then for all . Since is a fixed real number, must be a constant function.Since for all , they must both be equal to the same constant value.
Therefore, is necessarily true.
Similarly, let's fix at some value . Then for all . Since is a fixed real number, must be a constant function.Since for all , they must both be equal to the same constant value.
Therefore, is necessarily true.
- Option (A) is a special case where the constant is 0, but it's not necessarily true for all such functions.
- Option (C) contradicts the finding that they must be constants.
- Option (D) simplifies to , which is also only true if the constant is 0.
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