f(π₯) and g(π¦) are functions of x and y, respectively, and f(π₯) = g(π¦) forβ¦
2023
f(π₯) and g(π¦) are functions of x and y, respectively, and f(π₯) = g(π¦) for all real values of π₯ and π¦. 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)=constant
- C.
f(x)=constant and g(y) not constant
- D.
f(x)+g(y)=f(x)-g(y)
Attempted by 281 students.
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Correct answer: B
Key idea: the value cannot depend on x or y.
Fix any y0. For every x, f(x)=g(y0), so f takes the same value g(y0) for all x; therefore f is constant.
Fix any x0. For every y, g(y)=f(x0), so g takes the same value f(x0) for all y; therefore g is constant.
Conclusion: Both functions equal the same constant for all real x and y, so f(x)=g(y)=constant.