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))

Solution: (3)

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

f (x) – f (-x) = 0

1) f (x) = [ax + a-x] / [ax – a-x]

f (-x) = [a-x + ax] / [a-x – ax]

= [a2x + 1] / [- (a2x – 1)]

= [1 + a2x] / [1 – a2x]

2) f (x) = [ax + 1] / [ax – 1]

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

3) f (x) = x * [[ax – 1] / [ax + 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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