Question

Let $f: R \rightarrow R$ be defined as $f(x)=\sin \pi[x]+e^{-|x|}-e^x$, then $f(x)$ is [Note: $[ x ]$ denotes the largest integer less than or equal to $x$.]

• A. injective but not surjective.
• B. surjective but not injective.
• C. injective as well as surjective.
• D. neither injective nor surjective

Answer 1

Answer: (D)

We have $f ( x )= e ^{-| x |}- e ^{ x } \quad($ As $\sin \pi[ x ]=0 \forall x \in R )$
$=\begin{cases}0 & ; x<0 \\ e^{-x}-e^x & ; x \geq 0\end{cases}$
Clearly, $f ( x )$ is many-one and into function. So, $f ( x )$ is neither injective nor surjective.