State True or False: The period of tanx function is `2pi`

Answer

To determine whether the statement "The period of tan(x) function is \(2\pi\)" is true or false, we need to analyze the properties of the tangent function.

### Step-by-Step Solution:

1. **Understanding Periodicity**:
- A function is said to be periodic if it repeats its values at regular intervals. The smallest positive interval at which this occurs is called the period of the function.

2. **Tangent Function**:
- The tangent function, denoted as \(tan(x)\), is defined as \(tan(x) = \frac{sin(x)}{cos(x)}\).

3. **Identifying Asymptotes**:
- The tangent function has vertical asymptotes where the cosine function equals zero. These points occur at \(x = \frac{\pi}{2} + n\pi\) for any integer \(n\). This means that the function is undefined at these points.

4. **Graphing the Function**:
- When we graph \(tan(x)\), we observe that it approaches positive and negative infinity at the asymptotes. Between these asymptotes, the function takes on all real values.

5. **Finding the Period**:
- By examining the graph, we can see that the function repeats its pattern every \(\pi\) units. For example, if we look at the interval from \(0\) to \(\pi\), the function behaves in a certain way, and from \(\pi\) to \(2\pi\), it behaves similarly but shifted.

6. **Conclusion**:
- Since the tangent function repeats every \(\pi\) units, we conclude that the period of \(tan(x)\) is \(\pi\), not \(2\pi\).

7. **Final Answer**:
- Therefore, the statement "The period of tan(x) function is \(2\pi\)" is **False**.