The graph of the function $f(x) = -\tan(x)$ for $x$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ has some distinctive characteristics. Here's a detailed description:

Key Features of the Graph

1. Asymptotes:

   • The function $f(x) = -\tan(x)$ has vertical asymptotes at $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$. The graph approaches these lines but never touches them.

2. Behavior of the Function:

   • The tangent function, $\tan(x)$, increases from negative infinity to positive infinity as $x$ moves from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. However, since the function is negated, $f(x) = -\tan(x)$ will decrease from positive infinity to negative infinity as $x$ increases from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$.

3. Symmetry:

   • The graph is symmetric with respect to the origin, meaning it has odd symmetry. This means $f(-x) = -f(x)$.

4. Intersections:

   • The graph intersects the origin at the point $(0, 0)$ because $\tan(0) = 0$, so $f(0) = 0$.

Description of the Graph

• Starting at $x = -\frac{\pi}{2}$: The graph starts near positive infinity because $\tan(x)$ approaches negative infinity as $x$ approaches $-\frac{\pi}{2}$, and the negative sign inverts this.

• Passing through the Origin $(0, 0)$: The graph crosses the origin because $f(0) = -\tan(0) = 0$.

• Ending at $x = \frac{\pi}{2}$: The graph ends near negative infinity as $x$ approaches $\frac{\pi}{2}$ because $\tan(x)$ approaches positive infinity, and the negative sign inverts this.

If you want a visual graph, I can describe how to create it or provide more details. Would you like further details or a visual representation?

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Here are five related questions that might interest you:

1. How does the graph of $f(x) = \tan(x)$ differ from $f(x) = -\tan(x)$ in the same interval?
2. What happens to the graph of $f(x) = -\tan(x)$ if the interval is extended beyond $-\frac{\pi}{2}$ and $\frac{\pi}{2}$?
3. How would the graph change if we considered $f(x) = \tan(x)$ within a different interval, such as $\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$?
4. What is the effect of adding a constant to $f(x) = -\tan(x)$, such as $f(x) = -\tan(x) + 2$?
5. How do the graphs of the functions $f(x) = \sin(x)$, $f(x) = \cos(x)$, and $f(x) = \tan(x)$ compare within the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$?

Tip: When graphing trigonometric functions, paying attention to the behavior near asymptotes and zeros helps in understanding the overall shape and transformation effects on the function.