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.
Best Answer
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 $0$.