### 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 output values of $g$ when the input values of $x$ are between 0 and 2, including the endpoints.
To do that, I need to understand how $g$ behaves as a function, and what kind of transformation it applies to $x$.
I notice that $g$ is a composition of four copies of $f$, so applying $g$ to $x$ is the same as applying $f$ four times in a row.
That means I can simplify the expression for $g$ by applying $f$ one step at a time, starting from the innermost $f$.
The innermost $f$ takes $x$ and turns it into $3x-2$.
The next $f$ takes $3x-2$ and turns it into $3(3x-2)-2$, which simplifies to $9x-8$.
The next $f$ takes $9x-8$ and turns it into $3(9x-8)-2$, which simplifies to $27x-26$.
The last $f$ takes $27x-26$ and turns it into $3(27x-26)-2$, which simplifies to $81x-80$.
So, I can rewrite $g$ as $g(x)=81x-80$.
Now I have a simpler function to work with, and I can see that $g$ is a linear function with a slope of 81 and a y-intercept of -80.
This means that $g$ is increasing as $x$ increases, and it passes through the point $(0,-80)$.
To find the range of $g$, I need to find the minimum and maximum values of $g$ when $x$ is between 0 and 2.
The minimum value of $g$ occurs when $x$ is 0, since $g$ is increasing.
When $x=0$, $g(0)=81(0)-80=-80$.
The maximum value of $g$ occurs when $x$ is 2, since $g$ is increasing.
When $x=2$, $g(2)=81(2)-80=82$.
Therefore, the range of $g$ is the interval from -80 to 82, including the endpoints.