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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]."
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