Question

How do you determine if `f(x) = ln(x + sqrt(x^2 + 1))` is an even or odd function?

Answer 1

`f(x)` is an odd function.

Explanation:

• An even function is one for which `f(-x) = f(x)` for all `x` in the domain.

• An odd function is one for which `f(-x) = -f(x)` for all `x` in the domain.

Note that:

`(sqrt(x^2+1)+x)(sqrt(x^2+1)-x) = (x^2+1)-x^2 = 1`

So we find:

`f(-x) = ln(-x + sqrt(x^2+1))`

`= ln(1/(x+sqrt(x^2+1)))`

`= -ln(x + sqrt(x^2+1)) = -f(x)`

So `f(x)` is an odd function.