Discussion
\(f(x)\) 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 \(y\)?
Answer: B
Necessarily, \(f(x) = g(y) =\) constant.
Fix any value \(y_{0}\). Then \(f(x) = g(y_{0})\) for every real \(x\), so \(f\) takes the single value \(c = g(y_{0})\) everywhere: it is constant. Since \(f(x) = g(y)\) for every \(y\) as well, \(g(y) = c\) for all \(y\). Both functions are the same constant.
The constant need not be zero: \(f(x) = g(y) = 5\) satisfies the condition, so 'both equal 0' is not necessary. 'Neither is constant' is the opposite of what was just shown. The last option simplifies to \(2g(y) = 0\), i.e. \(g(y) = 0\), which again is only one special case.
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