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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.

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    $\begingroup$ Are you asked to show it is even/odd on a specific interval ? $\endgroup$
    – Ark
    May 20, 2021 at 12:42
  • $\begingroup$ @Ark Nope... It just mentions the function. I believe that was the catch here. $\endgroup$
    – Prajwal Tiwari
    May 20, 2021 at 12:53
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    $\begingroup$ 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. $\endgroup$
    – Ark
    May 20, 2021 at 12:57

1 Answer 1

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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 $0$.

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