The function f is defined by f x = 8 x . For what value of

Choice C is correct. It's given that $f\left(x\right)=8\sqrt{x}$. Substituting $48$ for $f\left(x\right)$ in this equation yields $48=8\sqrt{x}$. Dividing both sides of this equation by $8$ yields $6=\sqrt{x}$. This can be rewritten as $\sqrt{x}=6$. Squaring both sides of this equation yields $x=36$. Therefore, the value of $x$ for which $f\left(x\right)=48$ is $36$.

Choice A is incorrect. If $x=6$, $f\left(x\right)=8\sqrt{6}$, not $48$.

Choice B is incorrect. If $x=8$, $f\left(x\right)=8\sqrt{8}$, not $48$.

Choice D is incorrect. If $x=64$, $f\left(x\right)=8\sqrt{64}$, which is equivalent to $64$, not $48$.