f(x) and g(y) are functions of x and y, respectively, and f(x) - g(y) = 0 for all values of x and y. Which one of the following options is necessarily TRUE for x and y?
A. f(x) = 0 and g(y) = 0
B. f(x) = g(y) = constant
C. f(x) if constant and g(y) if constant
D. f(x) + g(y) = f(x) * g(y)
GATE 2023 · General Aptitude · Functions · medium
Answer: f(x) = g(y) = constant (Option B)
Rewrite the constraint: The constraint says f(x) = g(y) for ALL x and ALL y.
Apply the separation of variables argument: Similarly, choosing any fixed x = x0, g(y) = f(x0) for all y, so g(y) is also a constant equal to the same c.
Evaluate options: Option A says both are 0 -- true only if c = 0, not necessarily. Option B says both are equal to the same constant -- always true. Option C is incorrectly worded. Option D is not generally implied.