Which of the following is an even function?

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(C) function $f(x)$ is even if $f(-x) = f(x)$.Let us check option $C$: $f(x) = x \cdot \frac{a^x - 1}{a^x + 1}$.$f(-x) = (-x) \cdot \frac{a^{-x} - 1}{

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$f(x) = x \cdot \frac{a^x - 1}{a^x + 1}$

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(C) function $f(x)$ is even if $f(-x) = f(x)$.Let us check option $C$: $f(x) = x \cdot \frac{a^x - 1}{a^x + 1}$.$f(-x) = (-x) \cdot \frac{a^{-x} - 1}{a^{-x} + 1}$$f(-x) = (-x) \cdot \frac{\frac{1}{a^x} - 1}{\frac{1}{a^x} + 1}$$f(-x) = (-x) \cdot \frac{\frac{1 - a^x}{a^x}}{\frac{1 + a^x}{a^x}}$$f(-x) = (-x) \cdot \frac{-(a^x - 1)}{a^x + 1}$$f(-x) = x \cdot \frac{a^x - 1}{a^x + 1} = f(x)$.Since $f(-x) = f(x)$, the function in option $C$ is an even function.

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