Question

which statement describes the domain of the function $f(x) = \frac{3x}{4x^2 - 4}$?
all real numbers except $x = \frac{3}{4}$
all nonzero real numbers
all real numbers except $x = -1$ and $x = 1$
all real numbers

Explanation:

Step1: Identify the denominator

The function is \( f(x) = \frac{3x}{4x^2 - 4} \). The denominator is \( 4x^2 - 4 \).

Step2: Set denominator not equal to zero

To find the domain, we solve \( 4x^2 - 4
eq 0 \).
Factor the denominator: \( 4(x^2 - 1) = 4(x - 1)(x + 1)
eq 0 \).
This implies \( x - 1
eq 0 \) and \( x + 1
eq 0 \), so \( x
eq 1 \) and \( x
eq -1 \).

Answer:

all real numbers except \( x = -1 \) and \( x = 1 \)