The function $y = \tan{x}$ exhibits a variety of interesting properties when considering the domain $0 \leq x < \frac{\pi}{2}$. Let’s go over several key aspects:

1. Domain and Range:

• Domain: The function $y = \tan{x}$ is defined for all real values of $x$ except where $x = \frac{\pi}{2} + n\pi$ for any integer $n$, because the tangent function has vertical asymptotes at these points due to division by zero in the expression $\tan{x} = \frac{\sin{x}}{\cos{x}}$.

  • In the interval $0 \leq x < \frac{\pi}{2}$, $\tan{x}$ is defined for all $x$, with no points of discontinuity.

• Range: As $x$ increases from 0 to $\frac{\pi}{2}$, $\tan{x}$ takes all values from 0 to $+\infty$.

  • Specifically, $\lim_{x \to 0^+} \tan{x} = 0$ and $\lim_{x \to \frac{\pi}{2}^-} \tan{x} = +\infty$.

2. Increasing Function:

• The tangent function is increasing in the interval $0 \leq x < \frac{\pi}{2}$. This can be confirmed by noting that the derivative of $\tan{x}$, which is $\sec^2{x}$, is positive for all $x$ in this interval. Thus, as $x$ increases from 0 to $\frac{\pi}{2}$, the value of $\tan{x}$ also increases.

3. Asymptotic Behavior:

• As $x$ approaches $\frac{\pi}{2}$, $\tan{x}$ approaches infinity. This behavior is a result of the cosine function approaching zero as $x$ nears $\frac{\pi}{2}$, causing the ratio $\frac{\sin{x}}{\cos{x}}$ to grow without bound.

4. Symmetry:

• The tangent function is odd, meaning that $\tan(-x) = -\tan{x}$. This symmetry is important for understanding the behavior of the tangent function across different quadrants of the unit circle, but it doesn't directly impact the interval $0 \leq x < \frac{\pi}{2}$.

5. Continuity:

• $y = \tan{x}$ is continuous on the interval $0 \leq x < \frac{\pi}{2}$, except at the endpoint $x = \frac{\pi}{2}$, where it has an infinite discontinuity (vertical asymptote).

6. Graph Behavior:

• The graph of $y = \tan{x}$ starts at the origin (0,0) and increases without bound as $x$ approaches $\frac{\pi}{2}$. The graph has no maximum or minimum within this domain, only a vertical asymptote at $x = \frac{\pi}{2}$.
• The graph is smooth and monotonic, meaning it has no sudden jumps or oscillations.

7. Periodicity:

• While $\tan{x}$ is periodic with period $\pi$, within the interval $0 \leq x < \frac{\pi}{2}$, the function behaves as if it has no period, increasing without repeating.

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Would you like more details on any of these properties? Or do you have further questions about the behavior of the function?

Related Questions:

1. How do the derivative and second derivative of $y = \tan{x}$ behave for $0 \leq x < \frac{\pi}{2}$?
2. What is the relationship between $y = \tan{x}$ and its inverse function $y = \tan^{-1}{x}$ in the given interval?
3. Can you graph the function $y = \tan{x}$ over $0 \leq x < \frac{\pi}{2}$ and identify key features like intercepts and asymptotes?
4. What would the limit of $\tan{x}$ be if $x \to \frac{\pi}{2}$ from the left side?
5. How does $y = \tan{x}$ compare to other trigonometric functions like $y = \sin{x}$ or $y = \cos{x}$ in terms of growth within this interval?

Tip:

The behavior of tangent near its asymptotes can be better understood by observing the function’s graph or analyzing the corresponding values of sine and cosine as $x$ nears $\frac{\pi}{2}$.