Odd/Even functions and integration of them

[FaraDazed]

I was not sure where to post this here or in calculus, but seeing as the underlying basic principle of my question is regarding parity of functions I am posting it here, but feel free to move if needed.

Basically I am getting ready for a (intro to) QM exam and I still struggle with some basic concepts such as the parity of functions. I mean, I get the basic premise that if $f(-x)=f(x)$ then the function is even and if $f(-x)=-f(x)$ then it is odd, yet I still seem to struggle and things that are in my lecurers notes confuse me.

Essentially we are given a (wave) function which is essentially a cosine function $\sqrt{\frac{2}{a}} cos(\frac{3 \pi x}{a})$ to be precise, and looking at a past paper that has outline solutions, one question asks the parity of the function, I answered even since its a cos function and that is what the answer was. But then, it asks us to find the expectation value of position which requires squaring the given function and integrating over all space, but the question only has 2 marks so only quick sentence is needed, and the answer given in the solutions that it is zero because the integral is of an ODD function.

That is the bit that confuses me, how has it become an odd function, just simply by squaring it? It is still symmetric about the y-axis is it not? And if you put -x into either of the now two cosine functions, surely they both spit out the same number if x was inserted?

 

[blue_leaf77]

When you calculate the expectation value of position in one dimension, the integrand will be something like $x|\psi(x)|^2$. $x$ is an odd function, if furthermore $|\psi(x)|^2$ is even, then the product of them will be?

 

[FaraDazed]

When you calculate the expectation value of position in one dimension, the integrand will be something like $x|\psi(x)|^2$. $x$ is an odd function, if furthermore $|\psi(x)|^2$ is even, then the product of them will be?

Ah yeah. Sorry. I forgot about that pesky x, and was focusing only on the wavefunction and wavefunction squared thinking that I was missing something there!

Plus as a mature student, the time I learn about parity of numbers in general and multiplying them etc was about 16 years ago (probs longer)! Forgot that there were general rules about that! Gosh I feel stupid now haha.