### Instruction: Let $f(x)=3x-2$, and let $g(x)=f(f(f(f(x))))$.  If the domain of $g$ is $0\leq x\leq 2$, compute the range of $g$.### Response: So we want to compute $g(x)=f(f(f(f(x))))$.
Right, so let's start with $f(x)=3x-2$.
So $f(f(x))=f(3x-2)$.
And that means $f(f(x))=3(3x-2)-2=9x-6-2=9x-8$.
Right.  So $f(f(f(x)))=f(9x-8)$.
And that means $f(f(f(x)))=3(9x-8)-2=27x-24-2=27x-26$.
Ok and finally $g(x)=f(27x-26)$.
That means $g(x)=3(27x-26)-2=81x-78-2=81x-80$.
Now we want to find the range of $g$ given that the domain of $g$ is $0\leq x\leq 2$.
Since the domain is a closed interval, the range is also a closed interval.
So we just need to find the minimum and maximum values of $g$ on the interval $0\leq x\leq 2$.
The minimum value occurs when $x=0$ and the maximum value occurs when $x=2$.
So the minimum value is $g(0)=81(0)-80=-80$ and the maximum value is $g(2)=81(2)-80=162-80=82$.