Since, $f:R\to R$ such that $f\left(x\right)=3^{-x}$

Let $y_1$ and $y_2$ be two elements of $f\left(x\right)$ such that $y_1=y_2$

$\Rightarrow 3^{-x_1}=3^{-x_2} \Rightarrow x_1=x_2$

Since, if two images are equal, then their elements are equal, therefore it is one-one function.

Since, $f\left(x\right)$ is positive for every value of $x$, therefore $f\left(x\right)$ is into.

On differentiating w.r.t. $x$, we get $\frac{dy}{dx}=-3^{-x}\log 3<0$ for every value of $x$.

$\therefore$ It is decreasing function.

$\therefore$ Statement I and III are true.