If `y=(tanx)/(tan3x),` then

Options

A. `yin[1//3,3]`
B. `y!in[1//3,3]`
C. `yin[-3,-1//3]`
D. `yin[-3,-1//3]`

Answer

To solve the problem where \( y = \frac{\tan x}{\tan 3x} \), we will find the range of \( y \).

### Step 1: Use the formula for \( \tan 3x \)
We know that:
\[ \tan 3x = \frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x} \]
Substituting this into our expression for \( y \):
\[ y = \frac{\tan x}{\tan 3x} = \frac{\tan x}{\frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x}} = \frac{\tan x (1 - 3 \tan^2 x)}{3 \tan x - \tan^3 x} \]

### Step 2: Simplify the expression
Assuming \( \tan x \neq 0 \) (as it would make \( y \) undefined), we can simplify:
\[ y = \frac{1 - 3 \tan^2 x}{3 - \tan^2 x} \]
Let \( t = \tan^2 x \). Thus, we can rewrite \( y \) as:
\[ y = \frac{1 - 3t}{3 - t} \]

### Step 3: Analyze the expression
To find the range of \( y \), we need to analyze the expression:
\[ y = \frac{1 - 3t}{3 - t} \]
We will find the values of \( y \) as \( t \) varies from \( 0 \) to \( \infty \).

### Step 4: Find critical points
To find critical points, we can set the derivative of \( y \) with respect to \( t \) to zero. Using the quotient rule:
\[ \frac{dy}{dt} = \frac{(3 - t)(-3) - (1 - 3t)(-1)}{(3 - t)^2} \]
Setting the numerator equal to zero gives:
\[ -3(3 - t) + (1 - 3t) = 0 \]
Simplifying this:
\[ -9 + 3t + 1 - 3t = 0 \implies -8 = 0 \]
This indicates that there are no critical points in the domain of \( t \).

### Step 5: Evaluate limits
Now we evaluate the limits as \( t \) approaches \( 0 \) and \( \infty \):
- As \( t \to 0 \):
\[ y \to \frac{1 - 0}{3 - 0} = \frac{1}{3} \]
- As \( t \to \infty \):
\[ y \to \frac{-\infty}{-\infty} \text{ (indeterminate form, we need to analyze further)} \]
To analyze this limit:
\[ y \to \frac{-3t}{-t} = 3 \]

### Step 6: Determine the range of \( y \)
From our evaluations:
- As \( t \) approaches \( 0 \), \( y \) approaches \( \frac{1}{3} \).
- As \( t \) approaches \( \infty \), \( y \) approaches \( 3 \).

Thus, \( y \) can take values from \( -\infty \) to \( \frac{1}{3} \) and from \( 3 \) to \( \infty \), but it cannot take values between \( \frac{1}{3} \) and \( 3 \).

### Final Answer
The range of \( y \) is:
\[ y \in (-\infty, \frac{1}{3}) \cup (3, \infty) \]