To determine the domain of the function [tex]\( f(x) = \frac{2x}{3x^2 - 3} \)[/tex], we need to consider the values of [tex]\( x \)[/tex] for which the function is defined. The function is defined for all real numbers except where the denominator is zero, because division by zero is undefined.
Let's find the values of [tex]\( x \)[/tex] that make the denominator zero:
1. Set the denominator equal to zero:
[tex]\[
3x^2 - 3 = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
3x^2 - 3 = 0
\][/tex]
Add 3 to both sides:
[tex]\[
3x^2 = 3
\][/tex]
Divide both sides by 3:
[tex]\[
x^2 = 1
\][/tex]
3. Solve for [tex]\( x \)[/tex] by taking the square root of both sides:
[tex]\[
x = \pm 1
\][/tex]
This shows that the denominator of the function becomes zero at [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex]. Therefore, the function [tex]\( f(x) \)[/tex] is not defined at these points.
Thus, the domain of the function [tex]\( f(x) = \frac{2x}{3x^2 - 3} \)[/tex] is all real numbers except [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].
In interval notation, the domain is:
[tex]\[
(-\infty, -1) \cup (-1, 1) \cup (1, \infty)
\][/tex]
So, the domain of the function is all real numbers except [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex].
