Is $x\log\bigl(\cos(x)\bigr)$ an even or odd function?

Question

$$f(-x)=-x\log\bigl(\cos(-x)\bigr)=-x\log\bigl(\cos(x)\bigr)=-f(x)$$

So it seems an odd function and i've tried to draw the graph too.

But the suggested solution in my book says that it is an even function.

It is an odd function, which is evident from your calculation and the graph. — Sarvesh Ravichandran Iyer, Oct 09 '17 at 10:49
just a comment: you have marked this question as "real analysis", however the stated function is not defined in $\Bbb R$ for values of $x$ such that $\cos x<0$. — , Oct 09 '17 at 10:59
I've tagged this as real analysis because i found this exercise in a book of real analysis simply — Anne, Oct 09 '17 at 11:01
@Masacroso That's no problem as far as real-analysis conscerrns. For a function to be odd/even the identity $f(-x) = \pm f(x)$ only has to hold in the domain of $f$. The domain of $f$ is where $\cos(x)>0$ when speaking in a real analysis context. — skyking, Oct 09 '17 at 11:20
As you have written it it is an odd function, but you should of course take care to make sure that you've read/transcribed correctly. — skyking, Oct 09 '17 at 11:21

score 6 · Accepted Answer · answered Oct 09 '17 at 10:50

6

Yes, it is an odd function and your argument is correct.

answered Oct 09 '17 at 10:50

1 Answers1