Answer
To find the domain and range of the tangent function:
1. **Domain**: All real numbers except where \( \cos(x) = 0 \), which is at \( x = \frac{\pi}{2} + k\pi \) for any integer \( k \).
2. **Range**: All real numbers, since the tangent function can take any value.
So, the domain is all real numbers excluding \( x = \frac{\pi}{2} + k\pi \), and the range is all real numbers.
Solution
To find the domain and range of the tangent function, we can follow these steps:
### Step 1: Understanding the Tangent Function
The tangent function is defined as:
\[
\tan(x) = \frac{\sin(x)}{\cos(x)}
\]
This means that the tangent function is undefined wherever the cosine function is equal to zero, as division by zero is not allowed.
### Step 2: Finding the Domain
1. **Identify where \(\cos(x) = 0\)**:
The cosine function is equal to zero at:
\[
x = \frac{\pi}{2} + k\pi \quad \text{for } k \in \mathbb{Z}
\]
This means that the tangent function is undefined at these points.
2. **Express the Domain**:
The domain of the tangent function can be expressed as:
\[
\text{Domain: } x \in \mathbb{R} \setminus \left\{ \frac{\pi}{2} + k\pi \mid k \in \mathbb{Z} \right\}
\]
In simpler terms, the domain includes all real numbers except for the points where \(x\) is an odd multiple of \(\frac{\pi}{2}\).
### Step 3: Finding the Range
1. **Behavior of the Tangent Function**:
The tangent function can take on any real value. As \(x\) approaches the points where the function is undefined (i.e., \(\frac{\pi}{2} + k\pi\)), the value of \(\tan(x)\) approaches \(\infty\) or \(-\infty\).
2. **Express the Range**:
Therefore, the range of the tangent function is:
\[
\text{Range: } y \in \mathbb{R}
\]
This means that the tangent function can output any real number.
### Summary
- **Domain**: \(x \in \mathbb{R} \setminus \left\{ \frac{\pi}{2} + k\pi \mid k \in \mathbb{Z} \right\}\)
- **Range**: \(y \in \mathbb{R}\)
This concludes the process of finding the domain and range of the tangent function.
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