To determine the domain of the function [tex]\( f(x) = \frac{3x}{4x^2 - 4} \)[/tex], we first need to identify any values of [tex]\( x \)[/tex] that would make the denominator equal to zero, as division by zero is undefined.
Let's start by examining the denominator of the function:
[tex]\[ 4x^2 - 4 \][/tex]
For this to be zero, we need to solve the equation:
[tex]\[ 4x^2 - 4 = 0 \][/tex]
First, factor out the common term:
[tex]\[ 4(x^2 - 1) = 0 \][/tex]
Next, recognize that [tex]\( x^2 - 1 \)[/tex] is a difference of squares, which can be factored further:
[tex]\[ 4(x+1)(x-1) = 0 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[ (x+1)(x-1) = 0 \][/tex]
This gives us two solutions:
[tex]\[ x+1 = 0 \quad \text{or} \quad x-1 = 0 \][/tex]
[tex]\[ x = -1 \quad \text{or} \quad x = 1 \][/tex]
These are the values of [tex]\( x \)[/tex] for which the denominator is zero and thus the function [tex]\( f(x) \)[/tex] is undefined.
Therefore, the domain of the function [tex]\( f(x) = \frac{3x}{4x^2 - 4} \)[/tex] is all real numbers except [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex].
Thus, the correct statement describing the domain of the function is:
All real numbers except [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex].
