A function f(x) ` =(x^(2) -3x +2)/(x^(2)+ 2x -3)` is

Options

A. 1 is not in its domain
B. minimum at x=-3 and maximum at x=1
C. no point of maxima and minima
D. increasing in its domain

Answer

To solve the problem, we will analyze the function \( f(x) = \frac{x^2 - 3x + 2}{x^2 + 2x - 3} \) step by step.

### Step 1: Factor the Numerator and Denominator

First, we need to factor both the numerator and the denominator.

**Numerator:**
\[ x^2 - 3x + 2 = (x - 2)(x - 1) \]

**Denominator:**
\[ x^2 + 2x - 3 = (x + 3)(x - 1) \]

Thus, we can rewrite \( f(x) \):
\[ f(x) = \frac{(x - 2)(x - 1)}{(x + 3)(x - 1)} \]

### Step 2: Determine the Domain

The function is undefined where the denominator is zero. Setting the denominator to zero:
\[ (x + 3)(x - 1) = 0 \]
This gives us the points:
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]

Thus, the domain of \( f(x) \) is all real numbers except \( x = 1 \) and \( x = -3 \):
\[ \text{Domain: } x \in \mathbb{R}, x \neq 1, -3 \]

### Step 3: Simplify the Function

Since \( x - 1 \) is a common factor in both the numerator and denominator, we can simplify \( f(x) \):
\[ f(x) = \frac{x - 2}{x + 3} \quad \text{for } x \neq 1 \]

### Step 4: Find the Derivative

Next, we will find the derivative of \( f(x) \) using the quotient rule:
\[ f'(x) = \frac{(x + 3)(1) - (x - 2)(1)}{(x + 3)^2} \]
Simplifying the numerator:
\[ f'(x) = \frac{x + 3 - x + 2}{(x + 3)^2} = \frac{5}{(x + 3)^2} \]

### Step 5: Analyze the Derivative

The derivative \( f'(x) = \frac{5}{(x + 3)^2} \) is always positive for all \( x \) in the domain (since the square of any real number is positive). Therefore, \( f(x) \) is increasing on its entire domain.

### Step 6: Conclusion

Since \( f(x) \) is increasing throughout its domain and has no points of maxima or minima, we conclude:
1. The function is not defined at \( x = 1 \) and \( x = -3 \).
2. There are no points of maxima and minima.
3. The function is increasing on its entire domain.

### Final Answer

The correct options based on the analysis are:
- 1: Not in its domain (true)
- 3: No point of maxima and minima (true)
- 4: Increasing in its whole domain (true)