The range of the function `f(x)=(x^(2)-2)/(x^(2)-3)` is

Options

A. `(-oo,2/3] cup (1,oo)`
B. `y in R`
C. `y in (2/3, 1)`
D. None of these

Answer

To find the range of the function \( f(x) = \frac{x^2 - 2}{x^2 - 3} \), we will follow these steps:

### Step 1: Set the function equal to \( y \)
We start by letting \( y = f(x) \):
\[ y = \frac{x^2 - 2}{x^2 - 3} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ y(x^2 - 3) = x^2 - 2 \]
This simplifies to:
\[ yx^2 - 3y = x^2 - 2 \]

### Step 3: Rearrange the equation
Rearranging the equation, we get:
\[ yx^2 - x^2 = 3y - 2 \]
Factoring out \( x^2 \) from the left side:
\[ x^2(y - 1) = 3y - 2 \]

### Step 4: Solve for \( x^2 \)
Now, we can express \( x^2 \) in terms of \( y \):
\[ x^2 = \frac{3y - 2}{y - 1} \]

### Step 5: Determine the conditions for \( x^2 \)
Since \( x^2 \) must be non-negative (i.e., \( x^2 \geq 0 \)), we need:
\[ \frac{3y - 2}{y - 1} \geq 0 \]

### Step 6: Analyze the inequality
To analyze this inequality, we will find the critical points by setting the numerator and denominator to zero:
1. **Numerator**: \( 3y - 2 = 0 \) gives \( y = \frac{2}{3} \)
2. **Denominator**: \( y - 1 = 0 \) gives \( y = 1 \)

### Step 7: Test intervals around the critical points
We will test the sign of the expression \( \frac{3y - 2}{y - 1} \) in the intervals determined by the critical points:
- **Interval 1**: \( (-\infty, \frac{2}{3}) \)
- **Interval 2**: \( \left(\frac{2}{3}, 1\right) \)
- **Interval 3**: \( (1, \infty) \)

1. **For \( y < \frac{2}{3} \)** (e.g., \( y = 0 \)):
\[ \frac{3(0) - 2}{0 - 1} = \frac{-2}{-1} = 2 \quad (\text{positive}) \]

2. **For \( \frac{2}{3} < y < 1 \)** (e.g., \( y = 0.8 \)):
\[ \frac{3(0.8) - 2}{0.8 - 1} = \frac{2.4 - 2}{-0.2} = \frac{0.4}{-0.2} = -2 \quad (\text{negative}) \]

3. **For \( y > 1 \)** (e.g., \( y = 2 \)):
\[ \frac{3(2) - 2}{2 - 1} = \frac{6 - 2}{1} = 4 \quad (\text{positive}) \]

### Step 8: Compile the results
From our analysis, the expression \( \frac{3y - 2}{y - 1} \geq 0 \) holds in the intervals:
- \( (-\infty, \frac{2}{3}] \) (including \( \frac{2}{3} \))
- \( (1, \infty) \)

### Step 9: Write the final range
Thus, the range of the function \( f(x) \) is:
\[ \text{Range} = \left(-\infty, \frac{2}{3}\right] \cup (1, \infty) \]