# How do I approach functions that look like they're neither odd nor even but they actually are?

I came across the following function and was asked to determine if its odd or even or neither:

$$f(x)=x[x^2]+\frac{1}{\sqrt{1-x^2}}$$, where [.] is the greatest integer function.

I started with the general approach of finding $$f(-x)$$ which came out to be:

$$f(-x)=-x[x^2]+\frac{1}{\sqrt{1-x^2}}$$

Looking at it that way, it simply looks like its neither odd nor even.

But when I checked my book, it was given to be even. And plotting its graph on a graphing tool again revealed its symmetry about y-axis showing that its even. How can I tackle these kind of problems?

Its clear that its designed to lure someone into the trap of thinking its neither odd nor even. So I am thinking we should rearrange $$f(x)$$ and then find $$f(-x)$$ but I dont really know how.

• Are you asked to show it is even/odd on a specific interval ?
– Ark
May 20, 2021 at 12:42
• @Ark Nope... It just mentions the function. I believe that was the catch here. May 20, 2021 at 12:53
• Well the answer below is probably what the exercise wants you to do. The function is defined on $\mathbb{R} \backslash \{1,-1 \}$ on which it is neither odd nor even so I find it weird to ask without giving an interval.
– Ark
May 20, 2021 at 12:57

The trap is that the 2nd term, $$\frac{1}{\sqrt{1 - x^2}}$$ forces the function (i.e. the expression) to only be defined for $$-1 < x < 1$$. This implies that $$x^2 < 1.$$ This implies that throughout the domain of the function, the first term must evaluate to $$0$$.