Find the domain and the range of the function `y=f(x)`, where `f(x)` is given by
`tanx`

Answer

To find the domain and range of the function \( y = f(x) \), where \( f(x) = \tan x \), we can follow these steps:

### Step 1: Understanding the function
The function \( f(x) = \tan x \) is defined as the ratio of the sine and cosine functions:
\[ \tan x = \frac{\sin x}{\cos x} \]
This means that \( \tan x \) is undefined wherever \( \cos x = 0 \).

### Step 2: Finding the points where \( \tan x \) is undefined
The cosine function is zero at the following points:
\[ x = \frac{\pi}{2} + n\pi \quad \text{for } n \in \mathbb{Z} \]
This means that \( \tan x \) is undefined at:
- \( x = \frac{\pi}{2}, \frac{3\pi}{2}, -\frac{\pi}{2}, -\frac{3\pi}{2}, \ldots \)

### Step 3: Determining the domain
Since \( \tan x \) is undefined at these points, the domain of \( f(x) \) includes all real numbers except these points. In interval notation, the domain can be expressed as:
\[ \text{Domain} = \mathbb{R} \setminus \left\{ \frac{\pi}{2} + n\pi \mid n \in \mathbb{Z} \right\} \]
This means the domain consists of all real numbers except for the points where \( \tan x \) is undefined.

### Step 4: Analyzing the range
The function \( \tan x \) can take any real value as \( x \) approaches the points where it is undefined. As \( x \) approaches \( \frac{\pi}{2} \) from the left, \( \tan x \) approaches \( +\infty \), and as \( x \) approaches \( \frac{\pi}{2} \) from the right, \( \tan x \) approaches \( -\infty \). This behavior repeats for every interval of \( \pi \).

### Step 5: Determining the range
Since \( \tan x \) can take any value from \( -\infty \) to \( +\infty \), the range of \( f(x) \) is:
\[ \text{Range} = \mathbb{R} \]

### Final Answer
- **Domain**: \( \mathbb{R} \setminus \left\{ \frac{\pi}{2} + n\pi \mid n \in \mathbb{Z} \right\} \)
- **Range**: \( \mathbb{R} \)