May the floor function help you!

Algebra Level 4

Is $f(x)$ $=$ $(-1)^{\lfloor x \rfloor} \sqrt{\frac{1}{4}-(x-\frac{1}{2}-\lfloor x \rfloor)^2}$ an odd or even function?

Try not to use graph!

Notation: $\lfloor \cdot \rfloor$ denotes the floor function.

odd even We can't really tell! It depends on x!

1 solution

Ajinkya Shivashankar
Nov 23, 2016

if x is an integer, then the surd becomes 0 .

if x is not an integer, $\lfloor{-x}\rfloor=-\lfloor{x}\rfloor-1$

Putting this in above equation, we see that the surd remains unchanged.

Also if $\lfloor{x}\rfloor$ is odd, $\lfloor{-x}\rfloor$ is even and vice-versa.(For non-integers)

So value of $(-1)^{\lfloor{x}\rfloor}$ changes its sign.

Hence $f(-x)=-f(x)$

Nicely done.

Omar Sehlouli - 4 years, 6 months ago