### 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: I need to find the possible outputs of $g$ when the inputs are between 0 and 2.
Since $g$ is defined as applying $f$ four times to the input, it might help to simplify $g$ by finding a single expression for it.
To do that, I can start by plugging in $f(x)$ into itself, and see what I get.
That is, $f(f(x))=3f(x)-2=3(3x-2)-2=9x-8$.
This is another linear function, but with a larger slope and a different intercept.
I can repeat this process to find $f(f(f(x)))$ and $f(f(f(f(x))))$.
For $f(f(f(x)))$, I plug in $9x-8$ into $f$, and get $f(9x-8)=3(9x-8)-2=27x-26$.
For $f(f(f(f(x))))$, I plug in $27x-26$ into $f$, and get $f(27x-26)=3(27x-26)-2=81x-80$.
So, $g(x)=f(f(f(f(x))))=81x-80$.
This is a very steep linear function, and I can use it to find the range of $g$ by plugging in the endpoints of the domain.
When $x=0$, $g(0)=81(0)-80=-80$.
When $x=2$, $g(2)=81(2)-80=82$.