To determine the domain of the function [tex]\( f(x) = \frac{4x - 7}{4x + 20} \)[/tex], we need to identify any values of [tex]\( x \)[/tex] that would make the denominator equal to zero, since division by zero is undefined.
The denominator of this function is [tex]\( 4x + 20 \)[/tex]. We need to find the values of [tex]\( x \)[/tex] for which this expression equals zero:
[tex]\[
4x + 20 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
4x + 20 = 0 \\
4x = -20 \\
x = -5
\][/tex]
So, [tex]\( x = -5 \)[/tex] makes the denominator zero, thus [tex]\( x = -5 \)[/tex] is not included in the domain of the function.
The domain of [tex]\( f(x) \)[/tex] consists of all real numbers except [tex]\( x = -5 \)[/tex]. In interval notation, this is represented as the union of the intervals:
[tex]\[
(-\infty, -5) \cup (-5, \infty)
\][/tex]
Therefore, the domain of the function is:
[tex]\[
\boxed{(-\infty, -5) \cup (-5, \infty)}
\][/tex]
