To determine the domain of the function [tex]\( f(x) = \frac{3x}{4x^2 - 4} \)[/tex], we need to identify the values of [tex]\(x\)[/tex] that make the denominator zero.
The function is undefined if the denominator is zero, so we need to solve for [tex]\(x\)[/tex] in the equation [tex]\( 4x^2 - 4 = 0 \)[/tex].
Let's go through this step by step:
1. Set the denominator equal to zero:
[tex]\[
4x^2 - 4 = 0
\][/tex]
2. Factor the equation:
[tex]\[
4(x^2 - 1) = 0
\][/tex]
3. Simplify the equation inside the parenthesis:
[tex]\[
x^2 - 1 = 0
\][/tex]
4. Factor the quadratic expression:
[tex]\[
(x - 1)(x + 1) = 0
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
x - 1 = 0 \quad \text{or} \quad x + 1 = 0
\][/tex]
[tex]\[
x = 1 \quad \text{or} \quad x = -1
\][/tex]
Thus, the function [tex]\( f(x) = \frac{3x}{4x^2 - 4} \)[/tex] has vertical asymptotes (undefined points) at [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].
Therefore, the domain of the function is all real numbers except [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].
Among the given choices, the correct statement describing the domain of the function is:
All real numbers except [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex].
