To determine the domain of the function
[tex]\[ f(x) = \frac{4x - 9}{4x + 8}, \][/tex]
we need to find all the values of [tex]\( x \)[/tex] for which the function is defined.
A function is undefined when its denominator equals zero, as division by zero is not allowed. Therefore, we need to find the values of [tex]\( x \)[/tex] where the denominator [tex]\( 4x + 8 \)[/tex] is zero.
1. Set the denominator equal to zero:
[tex]\[ 4x + 8 = 0 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ 4x + 8 = 0 \][/tex]
[tex]\[ 4x = -8 \][/tex]
[tex]\[ x = -2 \][/tex]
The denominator is zero when [tex]\( x = -2 \)[/tex]. Hence, the function [tex]\( f(x) \)[/tex] is undefined at [tex]\( x = -2 \)[/tex].
The domain of [tex]\( f(x) \)[/tex] includes all real numbers except [tex]\( x = -2 \)[/tex]. In interval notation, we express the domain by excluding [tex]\( -2 \)[/tex] from the set of all real numbers.
Thus, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ (-\infty, -2) \cup (-2, \infty) \][/tex]
Therefore, the domain is
[tex]\[ \boxed{(-\infty, -2) \cup (-2, \infty)} \][/tex]
