Question
Suppose $f: \Re \to \Re$ is a real valued function defined on the whole real line. For each a) and b) determine if the statement is correct and justify your answer.
a) If $f(x)$ is even then $g(x)= \left \lfloor{f(x)}\right \rfloor $ even.
b) If $f(x)$ is odd then $g(x)= \left \lfloor{f(x)}\right \rfloor $ odd.
Things I know
I know that the floor function is the greatest integer less than the
value x.$f(x)=f(-x)$ if $f(x)$ is even and the graphs are symmetrical
about y=0$f(x)=-f(-x)$ if $f(x)$ is odd
I have tried to think of a way, but with the knowledge I dont know how to come to the conclusion if the statements are correct and to justify. But my knowledge is up to date with the school syllabus (what we have been taught already). How do I do this?
Best Answer
b) is incorrect. The easiest counter example is taking $f(x)=x$. Booldy's counter example is correct too because if $f(x)=\sin x$ then $g(x)=\lfloor\sin (-x)\rfloor=\lfloor -\sin x\rfloor$ but nothing ensures that $\lfloor -\sin x\rfloor=-\lfloor \sin x\rfloor$. Take $x=\dfrac{\pi}{4}$ to be convinced.
For a) just use the definition you have. Let $x\in\mathbb{R}$.$f$ is even so $f(-x)=f(x)$. Now use this while calculating $g(-x)$.