If $f: R \rightarrow R$,such that $f(x)=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$,then $f$ is

The function $f: R \rightarrow R$ is defined by $f(x)=3^{-x}$. Observe the following statements about it:
$I$. $f$ is one-one
$II$. $f$ is onto
$III$. $f$ is a decreasing function
Out of these, the true statements are:

What is the nature of the function $f(x) = 2 |x - 1| + 3 |x - 2|$ in the interval $(1, 2)$?

Check the injectivity and surjectivity of the following function $f : N \rightarrow N$ given by $f(x) = x^{3}$.

If $P(S)$ denotes the set of all subsets of a given set $S$,then the number of one-to-one functions from the set $S = \{ 1, 2, 3 \}$ to the set $P(S)$ is

Show that the function $f: N \rightarrow N$ given by $f(x) = 2x$ is one-one but not onto.