IDNLearn.com is the perfect place to get detailed and accurate answers to your questions. Whether your question is simple or complex, our community is here to provide detailed and trustworthy answers quickly and effectively.
Sagot :
To determine the behavior of the graph of the given function, let's analyze the provided table of [tex]\(x\)[/tex] and [tex]\(f(x)\)[/tex] values:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline x & -0.2 & -0.1 & 0 & 0.1 & 0.2 & 3.7 & 3.8 & 3.9 & 3.99 & 4 & 4.01 & 4.1 & 4.2 \\ \hline f(x) & -0.238 & -0.244 & \text{undefined} & -0.256 & -0.263 & -3.\overline{3} & -5 & -10 & -100 & \text{undefined} & 100 & 10 & 5 \\ \hline \end{array} \][/tex]
First, we'll examine where the function [tex]\(f(x)\)[/tex] is undefined, which happens at [tex]\(x = 0\)[/tex] and [tex]\(x = 4\)[/tex]. These points could correspond to vertical asymptotes or holes in the function's graph.
To differentiate between vertical asymptotes and holes, we examine the behavior of [tex]\(f(x)\)[/tex] as [tex]\(x\)[/tex] approaches these undefined points.
1. Behavior near [tex]\(x = 0\)[/tex]:
- For [tex]\(x\)[/tex] values near 0 ([tex]\(-0.2\)[/tex], [tex]\(-0.1\)[/tex], [tex]\(0.1\)[/tex], and [tex]\(0.2\)[/tex]), [tex]\(f(x)\)[/tex] has values [tex]\(-0.238\)[/tex], [tex]\(-0.244\)[/tex], [tex]\(-0.256\)[/tex], and [tex]\(-0.263\)[/tex] respectively.
- As [tex]\(x\)[/tex] approaches 0 from either direction ([tex]\(x \to 0^-\)[/tex] or [tex]\(x \to 0^+\)[/tex]), [tex]\(f(x)\)[/tex] does not tend towards [tex]\(\pm \infty\)[/tex]. Instead, [tex]\(f(x)\)[/tex] appears to vary smoothly.
2. Behavior near [tex]\(x = 4\)[/tex]:
- For [tex]\(x\)[/tex] values just below 4 ([tex]\(3.7\)[/tex], [tex]\(3.8\)[/tex], [tex]\(3.9\)[/tex], and [tex]\(3.99\)[/tex]), [tex]\(f(x)\)[/tex] values are [tex]\(-3.\overline{3}\)[/tex], [tex]\(-5\)[/tex], [tex]\(-10\)[/tex], and [tex]\(-100\)[/tex] respectively.
- For [tex]\(x\)[/tex] values just above 4 ([tex]\(4.01\)[/tex], [tex]\(4.1\)[/tex], and [tex]\(4.2\)[/tex]), [tex]\(f(x)\)[/tex] values are 100, 10, and 5 respectively.
- As [tex]\(x\)[/tex] approaches 4 from the left ([tex]\(x \to 4^-\)[/tex]), [tex]\(f(x)\)[/tex] decreases towards [tex]\(-\infty\)[/tex].
- As [tex]\(x\)[/tex] approaches 4 from the right ([tex]\(x \to 4^+\)[/tex]), [tex]\(f(x)\)[/tex] increases towards [tex]\(\infty\)[/tex].
Given the behaviors near [tex]\(x = 0\)[/tex] and [tex]\(x = 4\)[/tex]:
- At [tex]\(x = 0\)[/tex]: The function is undefined, but the values of [tex]\(f(x)\)[/tex] near 0 do not approach [tex]\(\pm \infty\)[/tex]. This behavior is indicative of a vertical asymptote.
- At [tex]\(x = 4\)[/tex]: The function is undefined, and the values of [tex]\(f(x)\)[/tex] nearby suggest [tex]\(f(x)\)[/tex] approaches [tex]\(\pm \infty\)[/tex]. This indicates a vertical asymptote rather than a hole.
Therefore, we can conclude that the function has vertical asymptotes at both [tex]\(x = 0\)[/tex] and [tex]\(x = 4\)[/tex].
The correct statement is:
"The function has vertical asymptotes when [tex]\(x = 0\)[/tex] and [tex]\(x = 4\)[/tex]."
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline x & -0.2 & -0.1 & 0 & 0.1 & 0.2 & 3.7 & 3.8 & 3.9 & 3.99 & 4 & 4.01 & 4.1 & 4.2 \\ \hline f(x) & -0.238 & -0.244 & \text{undefined} & -0.256 & -0.263 & -3.\overline{3} & -5 & -10 & -100 & \text{undefined} & 100 & 10 & 5 \\ \hline \end{array} \][/tex]
First, we'll examine where the function [tex]\(f(x)\)[/tex] is undefined, which happens at [tex]\(x = 0\)[/tex] and [tex]\(x = 4\)[/tex]. These points could correspond to vertical asymptotes or holes in the function's graph.
To differentiate between vertical asymptotes and holes, we examine the behavior of [tex]\(f(x)\)[/tex] as [tex]\(x\)[/tex] approaches these undefined points.
1. Behavior near [tex]\(x = 0\)[/tex]:
- For [tex]\(x\)[/tex] values near 0 ([tex]\(-0.2\)[/tex], [tex]\(-0.1\)[/tex], [tex]\(0.1\)[/tex], and [tex]\(0.2\)[/tex]), [tex]\(f(x)\)[/tex] has values [tex]\(-0.238\)[/tex], [tex]\(-0.244\)[/tex], [tex]\(-0.256\)[/tex], and [tex]\(-0.263\)[/tex] respectively.
- As [tex]\(x\)[/tex] approaches 0 from either direction ([tex]\(x \to 0^-\)[/tex] or [tex]\(x \to 0^+\)[/tex]), [tex]\(f(x)\)[/tex] does not tend towards [tex]\(\pm \infty\)[/tex]. Instead, [tex]\(f(x)\)[/tex] appears to vary smoothly.
2. Behavior near [tex]\(x = 4\)[/tex]:
- For [tex]\(x\)[/tex] values just below 4 ([tex]\(3.7\)[/tex], [tex]\(3.8\)[/tex], [tex]\(3.9\)[/tex], and [tex]\(3.99\)[/tex]), [tex]\(f(x)\)[/tex] values are [tex]\(-3.\overline{3}\)[/tex], [tex]\(-5\)[/tex], [tex]\(-10\)[/tex], and [tex]\(-100\)[/tex] respectively.
- For [tex]\(x\)[/tex] values just above 4 ([tex]\(4.01\)[/tex], [tex]\(4.1\)[/tex], and [tex]\(4.2\)[/tex]), [tex]\(f(x)\)[/tex] values are 100, 10, and 5 respectively.
- As [tex]\(x\)[/tex] approaches 4 from the left ([tex]\(x \to 4^-\)[/tex]), [tex]\(f(x)\)[/tex] decreases towards [tex]\(-\infty\)[/tex].
- As [tex]\(x\)[/tex] approaches 4 from the right ([tex]\(x \to 4^+\)[/tex]), [tex]\(f(x)\)[/tex] increases towards [tex]\(\infty\)[/tex].
Given the behaviors near [tex]\(x = 0\)[/tex] and [tex]\(x = 4\)[/tex]:
- At [tex]\(x = 0\)[/tex]: The function is undefined, but the values of [tex]\(f(x)\)[/tex] near 0 do not approach [tex]\(\pm \infty\)[/tex]. This behavior is indicative of a vertical asymptote.
- At [tex]\(x = 4\)[/tex]: The function is undefined, and the values of [tex]\(f(x)\)[/tex] nearby suggest [tex]\(f(x)\)[/tex] approaches [tex]\(\pm \infty\)[/tex]. This indicates a vertical asymptote rather than a hole.
Therefore, we can conclude that the function has vertical asymptotes at both [tex]\(x = 0\)[/tex] and [tex]\(x = 4\)[/tex].
The correct statement is:
"The function has vertical asymptotes when [tex]\(x = 0\)[/tex] and [tex]\(x = 4\)[/tex]."
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.
