Let f: R arrow R be defined as f(x)= sin π[x]+e-|x|-ex, then f(x) is [Note: [ x ] denotes the largest integer less than or equal to x.]

Question

Let f: R arrow R be defined as f(x)= sin π[x]+e-|x|-ex, 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

Tardigrade Admin · Accepted Answer

We have f ( x )= e -| x |- e x ( As sin π[ x ]=0 ∀ x ∈ R ) = begincases0 ; x<0 e-x-ex ; x ≥ 0 endcases Clearly, f ( x ) is many-one and into function. So, f ( x ) is neither injective nor surjective.

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

injective but not surjective.

surjective but not injective.

injective as well as surjective.

neither injective nor surjective

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

Q. Let f:R→R be defined as f(x)=sinπ[x]+e−∣x∣−ex, then f(x) is [Note: [x] denotes the largest integer less than or equal to x.]