To find the domain of the function [tex]\( h(x) = \frac{x-4}{x^3 - 81x} \)[/tex], we need to identify all values of [tex]\( x \)[/tex] for which the function is defined. Specifically, we need to find where the denominator is not zero, as division by zero is undefined.
Let's first consider the denominator of the function:
[tex]\[ d(x) = x^3 - 81x \][/tex]
To ensure the function is defined, we need to find where [tex]\( d(x) \neq 0 \)[/tex]. Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x^3 - 81x = 0 \][/tex]
Factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x(x^2 - 81) = 0 \][/tex]
This equation can be factored further. Note that [tex]\( x^2 - 81 \)[/tex] is a difference of squares:
[tex]\[ x(x - 9)(x + 9) = 0 \][/tex]
The solutions to this equation are:
[tex]\[ x = 0, x = 9, x = -9 \][/tex]
These are the values of [tex]\( x \)[/tex] that would make the denominator zero. Therefore, the function [tex]\( h(x) \)[/tex] is undefined at [tex]\( x = 0 \)[/tex], [tex]\( x = 9 \)[/tex], and [tex]\( x = -9 \)[/tex].
Consequently, the domain of [tex]\( h(x) \)[/tex] consists of all real numbers except [tex]\( x = 0 \)[/tex], [tex]\( x = 9 \)[/tex], and [tex]\( x = -9 \)[/tex]. Therefore, the correct notation for the domain is:
[tex]\[ \{x \mid x \neq -9, 0, 9\} \][/tex]
Thus, the correct answer is:
[tex]\[ \text{A. } \{x \mid x \neq -9,0,9\} \][/tex]
