Question

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?

• A. \( f(x) = 0 \) and \( g(y) = 0 \)
• B. \( f(x) = g(y) = \text{constant} \)
• C. \( f(x) \neq \text{constant} \) and \( g(y) \neq \text{constant} \)
• D. \( f(x) + g(y) = f(x) - g(y) \)

Hint:

When two functions are equal for all real values of their variables, they must be constant to maintain the equality.

Answer 1

The Correct Option is B

Since \( f(x) = g(y) \) for all real values of \( x \) and \( y \), this means that for any value of \( x \), the value of \( f(x) \) is equal to the value of \( g(y) \), implying that both functions must be constant to maintain equality across all values of \( x \) and \( y \). Thus, \( f(x) \) and \( g(y) \) must both be constants.
- (A) is incorrect because there is no necessity that \( f(x) = 0 \) and \( g(y) = 0 \) for all \( x \) and \( y \); both functions must just be constants.
- (C) is incorrect because the assumption that \( f(x) \) and \( g(y) \) are not constants contradicts the condition \( f(x) = g(y) \) for all \( x \) and \( y \).
- (D) is incorrect because \( f(x) + g(y) = f(x) - g(y) \) implies that \( g(y) \) must be 0, which is not necessary given the conditions.
Thus, the correct answer is (B): both \( f(x) \) and \( g(y) \) must be constants.