To determine the domain of the function [tex]\( f(x) = \frac{x-1}{x^2 - 2x - 15} \)[/tex], we need to identify the values of [tex]\( x \)[/tex] that make the denominator zero, as the function is undefined at these points. This requires solving the following equation:
[tex]\[ x^2 - 2x - 15 = 0 \][/tex]
The roots of this quadratic equation, which are the values of [tex]\( x \)[/tex] making the denominator zero, are [tex]\(\boxed{5.0}\)[/tex] and [tex]\(\boxed{-3.0}\)[/tex].
These roots show the points where the function [tex]\( f(x) \)[/tex] is undefined. Therefore, the domain of [tex]\( f(x) \)[/tex] includes all real numbers except for these values. We express this by saying [tex]\( x \neq 5 \)[/tex] and [tex]\( x \neq -3 \)[/tex].
Thus, the correct format to use is:
[tex]\[ x \neq M \][/tex]
The values that make the function undefined, and hence should be excluded from the domain, are [tex]\(5\)[/tex] and [tex]\(-3\)[/tex]. These should be filled into the respective blanks in the provided choice:
[tex]\[
\begin{array}{l}
x \neq 5, -3 \\
x \geq \\
x \leq
\end{array}
\][/tex]
Therefore, the filled boxes should look like this:
[tex]\[ x \neq 5, -3 \][/tex]
Confirming this, ensure that the other boxes are left blank. The domain of the function [tex]\( f(x) \)[/tex] is all real numbers except [tex]\( x = 5 \)[/tex] and [tex]\( x = -3 \)[/tex].
