### Instruction: Let $f(x)=\frac{1}{x-3}$.  Find the largest $x$ that is not in the domain of $g(x)=f(f(x))$.### Response: To find the domain of $g(x)$, I need to identify the values of $x$ that make $g(x)$ undefined or invalid.
One way that can happen is if $f(x)$ is undefined, which occurs when $x-3=0$, or $x=3$.
Another way that can happen is if $f(x)$ is equal to $3$, since that would make $f(f(x))$ undefined.
So I need to solve the equation $f(x)=3$, or $\frac{1}{x-3}=3$.
To do that, I can cross-multiply and get $x-3=\frac{1}{3}$, or $x=\frac{1}{3}+3=\frac{10}{3}$.
So $x=3$ and $x=\frac{10}{3}$ are both values that are not in the domain of $g(x)$.
The question asks for the largest such value, so I compare $3$ and $\frac{10}{3}$ and see that $\frac{10}{3}$ is larger.
Therefore, the largest $x$ that is not in the domain of $g(x)$ is $\frac{10}{3}$.