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Which of the following functions has a graph that is symmetric about the y-axis?

1. f(x) = sin[log(x + √(x² + 1))]
2. f(x) = (sec⁴x + cosec⁴x)/(x³ + x⁴ cot x)
3. f(x + y) = f(x) + f(y) for all x, y ∈ R
4. None of these

Correct answer: None of these

Solution

Option (a) sin[log(x+sqrt(x^2+1))] uses the odd function sinh^-1(x), so it is odd. Option (b) has an even numerator over an odd denominator (x^3 + x^4 cot x is odd), giving an odd function. Option (c) is Cauchy's equation forcing f(0)=0 and f(-x)=-f(x), odd. None is even, so answer is None of these.

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