Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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.
[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.
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
