Question

describing the behavior of rational functions
consider the following function:
$f(x) = \frac{2x}{3x^2 - 3}$
what is the domain of the function?
all real numbers
all nonzero real numbers
all real numbers except 2/3
all real numbers except -1 and 1

Explanation:

Step1: Find when denominator is zero

The function is \( f(x)=\frac{2x}{3x^{2}-3} \). The denominator cannot be zero. So solve \( 3x^{2}-3 = 0 \).

Step2: Solve the equation

Factor the denominator: \( 3(x^{2}-1)=0 \), then \( x^{2}-1 = 0 \), which factors to \( (x - 1)(x + 1)=0 \). So \( x=1 \) or \( x=-1 \). Thus, the domain is all real numbers except -1 and 1.

Answer:

all real numbers except -1 and 1