Question 3
Let $f(x)=\frac{3 x-6}{x^{2}+8 x+15}=\frac{(3 x-6)}{(x+5)(x+3)}$
Find:
1) the domain in interval notation
Note: Use -oo for $-\infty$, oo for $\infty$, U for union.
2) the $y$ intercept at the point
3) $x$ intercepts at the point(s)
4) Vertical asymptotes at $x=$
5) Horizontal asymptote at $y=$

Step 1 ：The domain of a function is the set of all possible input values (often the 'x' variable), which produce a valid output from a particular function. The only values that are not in the domain of this function are those that make the denominator equal to zero, because division by zero is undefined.

Step 2 ：We need to find the values of x that make the denominator equal to zero. The denominator of the function is \(x^{2} + 8x + 15\).

Step 3 ：The solutions to the equation \(x^{2}+8 x+15=0\) are \(x=-5\) and \(x=-3\). These are the values that make the denominator of the function equal to zero, and thus are not in the domain of the function.

Step 4 ：Therefore, the domain of the function is all real numbers except \(x=-5\) and \(x=-3\). In interval notation, this is \((-\infty, -5) \cup (-5, -3) \cup (-3, \infty)\).

Step 5 ：Final Answer: The domain of the function in interval notation is \(\boxed{(-\infty, -5) \cup (-5, -3) \cup (-3, \infty)}\).