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. This happens when the denominator is zero, as division by zero is undefined.
The denominator of the function is given by:
[tex]\[ 4x^2 - 4 \][/tex]
To find the values of [tex]\( x \)[/tex] that make the denominator zero, solve the equation:
[tex]\[ 4x^2 - 4 = 0 \][/tex]
First, factor out the common factor:
[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]
For the product to be zero, at least one of the factors must be zero:
[tex]\[ x + 1 = 0 \quad \text{or} \quad x - 1 = 0 \][/tex]
Solving these equations, we get:
[tex]\[ x = -1 \quad \text{or} \quad x = 1 \][/tex]
Thus, the function [tex]\( f(x) = \frac{3x}{4x^2 - 4} \)[/tex] is undefined at [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex]. Therefore, these values must be excluded from the domain.
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 is:
"all real numbers except [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex]".
