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] for which the function is undefined. The function is undefined wherever its denominator is zero because division by zero is undefined.
Let's examine the denominator of the function:
[tex]\[ 4x^2 - 4 \][/tex]
First, we set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 4x^2 - 4 = 0 \][/tex]
Next, we can factor out the common term:
[tex]\[ 4(x^2 - 1) = 0 \][/tex]
We recognize that [tex]\( x^2 - 1 \)[/tex] is a difference of squares:
[tex]\[ 4(x - 1)(x + 1) = 0 \][/tex]
Setting each factor equal to zero gives us:
[tex]\[ x - 1 = 0 \][/tex]
[tex]\[ x = 1 \][/tex]
[tex]\[ x + 1 = 0 \][/tex]
[tex]\[ x = -1 \][/tex]
So, the function [tex]\( f(x) = \frac{3x}{4x^2 - 4} \)[/tex] is undefined at [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].
This means the domain of the function is all real numbers except [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].
Therefore, the correct statement describing the domain of the function [tex]\( f(x) \)[/tex] is:
All real numbers except [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex].
