Problem 16. (1 point)
Consider the function
\[
f(x)=\log _{4}(15-6 x)
\]

Give the domain of the function $f(x)$ in interval notation.
Domain:
Give the range of the function $f(x)$ in interval notation.
Range:

Note: recall that you can express $\infty$ and $-\infty$ in an interval using Infinity and -Infinity respectively.
Note: You can earn partial credit on this problem.

Step 1 ：The domain of a function is the set of all possible input values (x-values) which will output real numbers. In this case, the function is a logarithm, and the argument of a logarithm (the part inside the log) must always be greater than zero. So, we need to find the values of x for which \(15-6x > 0\).

Step 2 ：The range of a logarithmic function is all real numbers, because a logarithm can output any real number depending on the input. However, this depends on the base of the logarithm. If the base is greater than 1, as it is in this case (base is 4), then the function is decreasing and the range is from negative infinity to positive infinity.

Step 3 ：Let's calculate the domain first. The domain of the function is all x such that \(x < \frac{5}{2}\). This can be written in interval notation as \((-∞, \frac{5}{2})\).

Step 4 ：Now let's consider the range. As I mentioned earlier, the range of a logarithmic function with a base greater than 1 is all real numbers. However, since the function is decreasing, the range is from negative infinity to positive infinity.

Step 5 ：Final Answer: The domain of the function \(f(x)\) is \(\boxed{(-∞, \frac{5}{2})}\) and the range of the function \(f(x)\) is \(\boxed{(-∞, ∞)}\).