Find the domain of the following real functions:
(i) `f(x)=(3x+5)/(x^(2)-9)` (ii) `f(x)=(2x-3)/(x^(2)+x-2)`
(iii) `f(x)=(x^(2)-2x+1)/(x^(2)-8x+12)` (iv) `f(x)=(x^(3)-8)/(x^(2)-1)`

Answer

To find the domain of the given functions, we need to identify the values of \( x \) for which the denominator is not equal to zero, since division by zero is undefined.

Let's solve each function step by step.

### (i) \( f(x) = \frac{3x + 5}{x^2 - 9} \)

1. **Identify the denominator**: The denominator is \( x^2 - 9 \).
2. **Set the denominator to zero**:
\[ x^2 - 9 = 0 \]
3. **Solve for \( x \)**:
\[ x^2 = 9 \implies x = 3 \text{ or } x = -3 \]
4. **Determine the domain**: The function is defined for all real numbers except \( x = 3 \) and \( x = -3 \).
\[ \text{Domain} = \mathbb{R} \setminus \{3, -3\} \]

### (ii) \( f(x) = \frac{2x - 3}{x^2 + x - 2} \)

1. **Identify the denominator**: The denominator is \( x^2 + x - 2 \).
2. **Set the denominator to zero**:
\[ x^2 + x - 2 = 0 \]
3. **Factor the quadratic**:
\[ (x - 1)(x + 2) = 0 \]
4. **Solve for \( x \)**:
\[ x = 1 \text{ or } x = -2 \]
5. **Determine the domain**: The function is defined for all real numbers except \( x = 1 \) and \( x = -2 \).
\[ \text{Domain} = \mathbb{R} \setminus \{1, -2\} \]

### (iii) \( f(x) = \frac{x^2 - 2x + 1}{x^2 - 8x + 12} \)

1. **Identify the denominator**: The denominator is \( x^2 - 8x + 12 \).
2. **Set the denominator to zero**:
\[ x^2 - 8x + 12 = 0 \]
3. **Factor the quadratic**:
\[ (x - 6)(x - 2) = 0 \]
4. **Solve for \( x \)**:
\[ x = 6 \text{ or } x = 2 \]
5. **Determine the domain**: The function is defined for all real numbers except \( x = 6 \) and \( x = 2 \).
\[ \text{Domain} = \mathbb{R} \setminus \{6, 2\} \]

### (iv) \( f(x) = \frac{x^3 - 8}{x^2 - 1} \)

1. **Identify the denominator**: The denominator is \( x^2 - 1 \).
2. **Set the denominator to zero**:
\[ x^2 - 1 = 0 \]
3. **Factor the quadratic**:
\[ (x - 1)(x + 1) = 0 \]
4. **Solve for \( x \)**:
\[ x = 1 \text{ or } x = -1 \]
5. **Determine the domain**: The function is defined for all real numbers except \( x = 1 \) and \( x = -1 \).
\[ \text{Domain} = \mathbb{R} \setminus \{1, -1\} \]

### Summary of Domains
- (i) \( \text{Domain} = \mathbb{R} \setminus \{3, -3\} \)
- (ii) \( \text{Domain} = \mathbb{R} \setminus \{1, -2\} \)
- (iii) \( \text{Domain} = \mathbb{R} \setminus \{6, 2\} \)
- (iv) \( \text{Domain} = \mathbb{R} \setminus \{1, -1\} \)