Certainly! Let's analyze the given function [tex]\( f(x) \)[/tex] using the provided table of values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-1000 & -1.9999982 \\
\hline
-0.1 & 1.78 \\
\hline
-0.0001 & 1.799998 \times 10^6 \\
\hline
0.0001 & 1.799998 \times 10^6 \\
\hline
0.1 & 1.78 \\
\hline
1000 & -1.9999982 \\
\hline
\end{array}
\][/tex]
Here are the key observations and insights we can draw from this table:
1. Asymptotic Behavior for Large [tex]\(|x|\)[/tex] Values:
- For [tex]\(x = -1000\)[/tex], [tex]\( f(x) = -1.9999982 \)[/tex].
- For [tex]\(x = 1000\)[/tex], [tex]\( f(x) = -1.9999982 \)[/tex].
This indicates that as [tex]\(x\)[/tex] becomes very large in the positive or negative direction, [tex]\( f(x) \)[/tex] approaches [tex]\(-2\)[/tex]. This suggests that the function has horizontal asymptotes at [tex]\( y = -2 \)[/tex] for large magnitudes of [tex]\( x \)[/tex].
2. Behavior Near Zero:
- For [tex]\( x = -0.0001 \)[/tex] and [tex]\( x = 0.0001 \)[/tex], [tex]\( f(x) \)[/tex] is extremely large, [tex]\( 1.799998 \times 10^6 \)[/tex].
This suggests that as [tex]\( x \)[/tex] approaches zero from either the positive or negative side, the function value [tex]\( f(x) \)[/tex] increases dramatically. The function seems to exhibit a vertical asymptote near [tex]\( x = 0 \)[/tex].
3. Symmetry:
- For [tex]\( x = -0.1 \)[/tex] and [tex]\( x = 0.1 \)[/tex], [tex]\( f(x) = 1.78 \)[/tex], which are identical.
- Similarly, symmetry is observed for [tex]\(x = -0.0001\)[/tex] and [tex]\( x = 0.0001 \)[/tex] with identical [tex]\( f(x) \)[/tex] values.
The function appears to be symmetric around the y-axis. This suggests that [tex]\( f(x) \)[/tex] might be an even function, satisfying [tex]\( f(x) = f(-x) \)[/tex].
4. Intermediate Values:
- For [tex]\( x = -0.1 \)[/tex], [tex]\( f(x) = 1.78 \)[/tex].
- For [tex]\( x = 0.1 \)[/tex], [tex]\( f(x) = 1.78 \)[/tex].
These values indicate that between 0 and 0.1 (either positive or negative), the function decreases from a very large value (near-zero) down to 1.78.
In summary, the function [tex]\( f(x) \)[/tex]:
- Approaches [tex]\(-2\)[/tex] for large positive and negative values of [tex]\( x \)[/tex].
- Exhibits a dramatic increase in value as [tex]\( x \)[/tex] approaches zero from either side.
- Shows symmetry around the y-axis, indicating it may be an even function.
- Attains similar values for [tex]\( x = \pm 0.1 \)[/tex], indicating behavior consistent with a smooth transition between the observed values.
Thus, the function [tex]\( f(x) \)[/tex] demonstrates interesting behavior characteristic of a function with both horizontal and vertical asymptotes and symmetry with respect to the y-axis.
