If `x in [1,2]` , then find the range of `f(x)=tanx`

Explanation

To find the range of the function \( f(x) = \tan x \) for \( x \in [1, 2] \), we will follow these steps:

### Step 1: Identify the interval and key points
We know that \( x \) is in the interval \([1, 2]\). We also need to consider the behavior of the tangent function within this interval. The tangent function has a vertical asymptote at \( x = \frac{\pi}{2} \) (approximately 1.57), which is relevant since it lies within our interval.

### Step 2: Calculate \( f(x) \) at the endpoints
We will evaluate the function at the endpoints of the interval:
- \( f(1) = \tan(1) \)
- \( f(2) = \tan(2) \)

Using a calculator or a trigonometric table:
- \( \tan(1) \approx 1.5574 \)
- \( \tan(2) \approx -2.1850 \)

### Step 3: Analyze the behavior of \( f(x) \) in the interval
Since \( \tan x \) is continuous and increases from \( x = 1 \) to \( x = \frac{\pi}{2} \), it will reach a maximum value at \( x = \frac{\pi}{2} \) (which is not included in the interval). As \( x \) approaches \( \frac{\pi}{2} \), \( \tan x \) approaches \( +\infty \).

After \( x = \frac{\pi}{2} \), the function will start decreasing again towards \( -\infty \) as \( x \) approaches \( 2 \).

### Step 4: Determine the range
From the analysis:
- As \( x \) increases from \( 1 \) to \( \frac{\pi}{2} \), \( f(x) \) increases from \( \tan(1) \) to \( +\infty \).
- As \( x \) increases from \( \frac{\pi}{2} \) to \( 2 \), \( f(x) \) decreases from \( -\infty \) to \( \tan(2) \).

Thus, the range of \( f(x) \) on the interval \( [1, 2] \) is:
\[ (-\infty, \tan(2)] \cup [\tan(1), +\infty) \]

### Final Answer
The range of \( f(x) = \tan x \) for \( x \in [1, 2] \) is:
\[ (-\infty, \tan(2)] \cup [\tan(1), +\infty) \]