1. **State the problem:** Given a function $f(x)$ with domain $[1,9]$ and range $[-3,6]$, find the domain and range of the function $g(x) = 3f(x)$.\n\n2. **Recall the domain and range definitions:**\n- The domain of a function is the set of all possible input values $x$.\n- The range is the set of all possible output values $f(x)$.\n\n3. **Domain of $g(x)$:**\nSince $g(x) = 3f(x)$ is defined by multiplying the output of $f(x)$ by 3, the input values $x$ for $g$ are the same as for $f$.\nTherefore, the domain of $g$ is the same as the domain of $f$: $$[1,9]$$\n\n4. **Range of $g(x)$:**\nThe range of $f$ is $[-3,6]$. Multiplying each value in the range by 3 scales the range by 3.\nCalculate the new range:\n$$3 \times (-3) = -9$$\n$$3 \times 6 = 18$$\nSo the range of $g$ is $$[-9,18]$$\n\n**Final answer:**\n- Domain of $g(x)$ is $[1,9]$.\n- Range of $g(x)$ is $[-9,18]$.