To determine the domain of the function
[tex]\[ f(x) = \frac{x+6}{(x-7)(x+5)} \][/tex]
we need to consider the values for which the function is defined. Specifically, the function is undefined when the denominator is zero because division by zero is undefined.
First, let's identify the values of [tex]\( x \)[/tex] that make the denominator zero:
[tex]\[ (x-7)(x+5) = 0 \][/tex]
We solve this equation by setting each factor in the denominator equal to zero:
1. [tex]\( x - 7 = 0 \)[/tex]
[tex]\[
x = 7
\][/tex]
2. [tex]\( x + 5 = 0 \)[/tex]
[tex]\[
x = -5
\][/tex]
Thus, the function [tex]\( f(x) \)[/tex] is undefined at [tex]\( x = 7 \)[/tex] and [tex]\( x = -5 \)[/tex]. Therefore, the domain of the function includes all real numbers except [tex]\( x = 7 \)[/tex] and [tex]\( x = -5 \)[/tex].
So the correct answer is:
C. all real numbers except [tex]\(-5\)[/tex] and [tex]\(7\)[/tex]
