Question

Determine whether function; $f\left(x\right)=(-1{)}^{\left[x\right]}$ is even, odd or neither of the two (where $[\cdot ]$ denotes the greatest integer function).

• A. $f$ is an even function when $x\in Z$, $f$ is an odd function when $x\notin Z$
• B. $f$ is an even function when $x\notin Z$ , $f$ is an odd function when $x\in Z$
• C. $f$ is an even function for all integers
• D. Neither odd nor even

Answer 1

Answer: (A), (C), (D)

The correct options are
A $f$ is an even function when $x\in Z$, $f$ is an odd function when $x\notin Z$
C $f$ is an even function for all integers
D Neither odd nor even

Let $f:R\to R$ be a function defined as $f\left(x\right)={\left(-1\right)}^{\left[x\right]}$.
When $x$ is an integer $\left[x\right]=x$, hence $f\left(x\right)={\left(-1\right)}^{\left[x\right]}={\left(-1\right)}^x$, which is an even function.
If $x$ is a positive real number which is not an integer, then $\left[x\right]$ is the integer part of the given number, say $a$.
Therefore $f\left(x\right)=\pm 1$ based on $a$ is even or odd.
If $x$ is a negative real number which is not an integer, then $\left[x\right]=a-1$, where $a$ is the integer part of the given number.
Therefore $f\left(x\right)=\pm 1$ based on $a$ is odd or even.
Hence $f\left(x\right)=-f\left(-x\right)$ when $x\notin Z$