Odd or Even Function: Logarithmic Equation Analysis for Homework

[Saitama]

Homework Statement

Determine whether the following function is odd or even or none:
$$f(x)=\log \left(x+\sqrt{1+x^2} \right)$$

Homework Equations

The Attempt at a Solution

For an even function, $f(-x)=f(x)$ and for an odd function $f(-x)=-f(x)$.
Replacing x with -x in the given function,
$$f(-x)=\log\left(-x+\sqrt{1+x^2}\right)$$
I don't see it being equal to f(x) or -f(x) so it should be neither odd nor even but the answer key states it is an odd function.

Any help is appreciated. Thanks!

 

[dirk_mec1]

hint: simplify the argument.

 

[Saitama]

hint: simplify the argument.

The only thing I can think of is multiplying and dividing by $\sqrt{1+x^2}+x$, do you ask me this?

 

[LCKurtz]

The only thing I can think of is multiplying and dividing by $\sqrt{1+x^2}+x$, do you ask me this?

Why don't you just try it and see?

 

[Saitama]

Why don't you just try it and see?

Woops, just tried that and it does come out to be -f(x). Thanks dirk_mec1!

 

[verty]

I like this question, very nice it is.

Hint: compare f(x) and f(-x), what do you see inside the logs? They are begging to have something done to them...

You've solved it now I see, so I'll just mention it. They are conjugate expressions that deserve to be multiplied together.

 

[Saitama]

You've solved it now I see, so I'll just mention it. They are conjugate expressions that deserve to be multiplied together.

Multiplied? Or should I add them?

 

[haruspex]

Multiplied? Or should I add them?

Multiply the expressions inside the logs, which is the same as adding the logs.
Going back to the problem as given, the most natural thing to try is f(x)+f(-x). As soon as you see the form log(..)+log(..), the next step should be automatic.