### 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: The range of $g$ is the set of all possible outputs of $g$, given that the inputs are restricted to the domain $0\leq x\leq 2$.
To find the range of $g$, I need to apply $f$ four times to any input $x$ in the domain and see what values I get.
I can start by applying $f$ once to $x$.
I get $f(x)=3x-2$.
Then I apply $f$ again to $f(x)$.
I get $f(f(x))=3(3x-2)-2=9x-8$.
I repeat this process two more times.
I get $f(f(f(x)))=3(9x-8)-2=27x-26$ and $f(f(f(f(x))))=3(27x-26)-2=81x-80$.
So $g(x)=f(f(f(f(x))))=81x-80$.
Now I need to see what values of $g(x)$ I get when $x$ varies from $0$ to $2$.
When $x=0$, I get $g(0)=81(0)-80=-80$.
This is the smallest possible value of $g(x)$, since $g(x)$ is an increasing function.
When $x=2$, I get $g(2)=81(2)-80=82$.
This is the largest possible value of $g(x)$, since $g(x)$ is an increasing function.
Therefore, the range of $g$ is the set of all numbers between $-80$ and $82$, inclusive.
I can write this using interval notation as $[-80,82]$.