If $f: R \rightarrow R$ is defined by $f(x) = x + 2|x + 1| + 2|x - 1|$,then the element in the co-domain,which has a unique pre-image in the domain is

Let $f : R \rightarrow R$ be defined as $f(x) = 3^{-|x|} - 3^x + \operatorname{sgn}(e^{-x}) + 2$ (where $\operatorname{sgn}(x)$ denotes the signum function of $x$). Then which one of the following is correct?

If $n(A) = 5$ and $n(B) = 8$,how many possible functions can be defined from $A$ to $B$?

If a function $f: R \rightarrow R$ is defined by $f(x)=x^3-x$,then $f$ is

Let $R = \{ a, b, c, d, e \}$ and $S = \{1, 2, 3, 4\}$. The total number of onto functions $f: R \rightarrow S$ such that $f(a) \neq 1$ is equal to $.............$.

Let $f : R \rightarrow R$ be a function such that $f(x) = \frac{x^2+2x+1}{x^2+1}$. Then