To find the domain of the rational function [tex]\( R(x) = \frac{8(x^2 - x - 56)}{9(x^2 - 64)} \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which the function is defined. Specifically, we need to find the values of [tex]\( x \)[/tex] that do not make the denominator equal to zero because division by zero is undefined.
First, consider the denominator of the rational function:
[tex]\[ 9(x^2 - 64) \][/tex]
We need to find the values of [tex]\( x \)[/tex] that make this expression equal to zero:
[tex]\[ 9(x^2 - 64) = 0 \][/tex]
To solve this equation, we start by simplifying it:
[tex]\[ x^2 - 64 = 0 \][/tex]
Next, we factor the quadratic expression:
[tex]\[ (x - 8)(x + 8) = 0 \][/tex]
Setting each factor equal to zero gives us:
[tex]\[ x - 8 = 0 \quad \text{or} \quad x + 8 = 0 \][/tex]
[tex]\[ x = 8 \quad \text{or} \quad x = -8 \][/tex]
Thus, the values of [tex]\( x \)[/tex] that make the denominator zero are [tex]\( x = 8 \)[/tex] and [tex]\( x = -8 \)[/tex]. These values must be excluded from the domain of [tex]\( R(x) \)[/tex].
The domain of [tex]\( R(x) \)[/tex] is all real numbers except where the denominator is zero. Therefore, we exclude [tex]\( x = 8 \)[/tex] and [tex]\( x = -8 \)[/tex].
In set notation, the domain of [tex]\( R(x) \)[/tex] can be expressed as:
[tex]\[ \{ x \mid x \neq 8 \text{ and } x \neq -8 \} \][/tex]
So, the correct choice is:
A. The domain of [tex]\( R(x) \)[/tex] is [tex]\(\{ x \mid x \neq -8, x \neq 8 \}\)[/tex].
