Question
What are key feature of a tangent parent function
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Answered by
GPT-4o mini
The tangent parent function is represented by \( f(x) = \tan(x) \). Here are the key features of the tangent function:
1. **Domain**: The tangent function is undefined at odd multiples of \( \frac{\pi}{2} \) (or \( 90^\circ \)), which means the domain can be expressed as:
\[
\text{Domain: } x \in \mathbb{R} \setminus \left\{ \frac{\pi}{2} + k\pi : k \in \mathbb{Z} \right\}
\]
In simpler terms, \( x \) can take any real number value except where \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \) etc.
2. **Range**: The range of the tangent function is all real numbers:
\[
\text{Range: } (-\infty, +\infty)
\]
3. **Periodicity**: The tangent function has a period of \( \pi \). This means the function repeats every \( \pi \) radians:
\[
f(x + \pi) = f(x)
\]
4. **Asymptotes**: There are vertical asymptotes at \( x = \frac{\pi}{2} + k\pi \) (where the function is undefined and approaches infinity):
5. **Zeros**: The tangent function crosses the x-axis at integer multiples of \( \pi \):
\[
\text{Zeros: } x = k\pi \, (k \in \mathbb{Z})
\]
6. **Symmetry**: The tangent function is an odd function, meaning it has rotational symmetry about the origin:
\[
f(-x) = -f(x)
\]
7. **Shape**: The graph of the tangent function has a characteristic shape that features increasing segments and sharp transitions at the asymptotes. As \( x \) approaches the vertical asymptotes, \( f(x) \) tends towards \( +\infty \) or \( -\infty \).
8. **Graph**: The graph of the tangent function is continuous between its asymptotes and starts at the origin (0,0) in each period, increases towards positive infinity, and then decreases back down towards negative infinity before approaching the next vertical asymptote.
Understanding these key features helps in analyzing the behavior of the tangent function in different contexts, such as in calculus or trigonometric applications.
1. **Domain**: The tangent function is undefined at odd multiples of \( \frac{\pi}{2} \) (or \( 90^\circ \)), which means the domain can be expressed as:
\[
\text{Domain: } x \in \mathbb{R} \setminus \left\{ \frac{\pi}{2} + k\pi : k \in \mathbb{Z} \right\}
\]
In simpler terms, \( x \) can take any real number value except where \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \) etc.
2. **Range**: The range of the tangent function is all real numbers:
\[
\text{Range: } (-\infty, +\infty)
\]
3. **Periodicity**: The tangent function has a period of \( \pi \). This means the function repeats every \( \pi \) radians:
\[
f(x + \pi) = f(x)
\]
4. **Asymptotes**: There are vertical asymptotes at \( x = \frac{\pi}{2} + k\pi \) (where the function is undefined and approaches infinity):
5. **Zeros**: The tangent function crosses the x-axis at integer multiples of \( \pi \):
\[
\text{Zeros: } x = k\pi \, (k \in \mathbb{Z})
\]
6. **Symmetry**: The tangent function is an odd function, meaning it has rotational symmetry about the origin:
\[
f(-x) = -f(x)
\]
7. **Shape**: The graph of the tangent function has a characteristic shape that features increasing segments and sharp transitions at the asymptotes. As \( x \) approaches the vertical asymptotes, \( f(x) \) tends towards \( +\infty \) or \( -\infty \).
8. **Graph**: The graph of the tangent function is continuous between its asymptotes and starts at the origin (0,0) in each period, increases towards positive infinity, and then decreases back down towards negative infinity before approaching the next vertical asymptote.
Understanding these key features helps in analyzing the behavior of the tangent function in different contexts, such as in calculus or trigonometric applications.