The even function is

Solution: (3)

Even function: f (-x) = f (x)

f (x) – f (-x) = 0

1) f (x) = [a^x + a^-x] / [a^x – a^-x]

f (-x) = [a^-x + a^x] / [a^-x – a^x]

= [a^2x + 1] / [- (a^2x – 1)]

= [1 + a^2x] / [1 – a^2x]

2) f (x) = [a^x + 1] / [a^x – 1]

f (-x) = [a^-x + 1] / [a^-x – 1] = [a^x + 1] / [-a^x + 1]

3) f (x) = x * [[a^x – 1] / [a^x + 1]]

f (-x) = -x * [[a^-x – 1] / [a^-x + 1]]

f (x) – f (-x) = 0

It is an even function.

4) f (x) = log₂ (x + √x² + 1)

f (-x) = log₂ [-x + √x² + 1]

f (x) – f (-x) = log₂ [2x² + 1 + 2x √x² – 1]

It is not an even function.

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The even function is (1) f (x) = [a^x + a^-x] / [a^x - a^-x] (2) f (x) = [a^x + 1] / [a^x - 1] (3) f (x) = x * [[a^x - 1] / [a^x + 1]] (4) f (x) = log_2 (x + root(x^2 + 1))