To determine the domain of the function [tex]\( f(x) = \frac{2x}{3x^2 - 3} \)[/tex], let's examine the denominator [tex]\( 3x^2 - 3 \)[/tex]. The function is defined wherever the denominator is not equal to zero. So, we need to find the values of [tex]\( x \)[/tex] for which [tex]\( 3x^2 - 3 = 0 \)[/tex]:
[tex]\[ 3x^2 - 3 = 0 \][/tex]
[tex]\[ 3x^2 = 3 \][/tex]
[tex]\[ x^2 = 1 \][/tex]
[tex]\[ x = \pm 1 \][/tex]
Therefore, the function is undefined at [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex]. The domain of [tex]\( f(x) \)[/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]
Next, let's describe the end behavior of the function [tex]\( f(x) = \frac{2x}{3x^2 - 3} \)[/tex]. For the end behavior, we examine the limits of the function as [tex]\( x \)[/tex] approaches positive infinity ([tex]\(+\infty\)[/tex]) and negative infinity ([tex]\(-\infty\)[/tex]):
1. As [tex]\( x \)[/tex] approaches positive infinity ([tex]\(+\infty\)[/tex]):
[tex]\[
\lim_{x \to \infty} \frac{2x}{3x^2 - 3}
\][/tex]
The degree of the polynomial in the numerator is 1, and the degree of the polynomial in the denominator is 2. Since the degree of the denominator is higher, the function approaches 0.
[tex]\[
\lim_{x \to \infty} f(x) = 0
\][/tex]
2. As [tex]\( x \)[/tex] approaches negative infinity ([tex]\(-\infty\)[/tex]):
[tex]\[
\lim_{x \to -\infty} \frac{2x}{3x^2 - 3}
\][/tex]
Similarly, the degree of the polynomial in the numerator is 1, and the degree of the polynomial in the denominator is 2. As in the positive infinity case, the function approaches 0.
[tex]\[
\lim_{x \to -\infty} f(x) = 0
\][/tex]
Given these results, we can answer the question about the end behavior of [tex]\( f(x) \)[/tex]:
- The graph approaches 0 as [tex]\( x \)[/tex] approaches infinity.
- The graph approaches 0 as [tex]\( x \)[/tex] approaches negative infinity.
The other statements do not accurately describe the end behavior of the function given the behavior we analyzed above. Therefore, the correct descriptions are:
- The graph approaches 0 as [tex]\( x \)[/tex] approaches infinity.
- The graph approaches 0 as [tex]\( x \)[/tex] approaches negative infinity.
