To determine the domain of the function [tex]\( f(x) = \frac{5}{1 - x^2} \)[/tex], we need to identify the values of [tex]\( x \)[/tex] for which the function is defined.
A function is undefined when the denominator is zero because division by zero is not possible. Therefore, we first need to find the values of [tex]\( x \)[/tex] that make the denominator zero.
Here, the denominator of our function is [tex]\( 1 - x^2 \)[/tex]. We set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 1 - x^2 = 0 \][/tex]
[tex]\[ x^2 = 1 \][/tex]
[tex]\[ x = \pm1 \][/tex]
This means that the function [tex]\( f(x) = \frac{5}{1 - x^2} \)[/tex] is undefined at [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex]. Therefore, these values must be excluded from the domain.
Thus, the domain of the function [tex]\( f(x) \)[/tex] includes all real numbers except [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].
In interval notation, this is expressed as:
[tex]\[ (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \][/tex]
So, the domain of the function is [tex]\(\boxed{(-\infty, -1) \cup (-1, 1) \cup (1, \infty)}\)[/tex].
