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Choice D is correct. It's given that for $x>0$, $f\left(x\right)$ is equal to $201\%$ of $x$. This is equivalent to $f\left(x\right)=\frac{201}{100}x$, or $f\left(x\right)=2.01x$, for $x>0$. This function indicates that as $x$ increases, $f\left(x\right)$ also increases, which means $f$ is an increasing function. Furthermore,  $f\left(x\right)$ increases at a constant rate of $2.01$ for each increase of $x$ by $1$. A function with a constant rate of change is linear. Thus, the function $f$ can be described as an increasing linear function.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect. This could describe the function $f\left(x\right)={\left(2.01\right)}^x$, where $f\left(x\right)$ is equal to $201\%$ of $f\left(x-1\right)$, not $x$, for $x>0$.