Find the domain of each of the following functions: `f(x)=(x^(3)-5x+3)/(x^(2)-1)`

Explanation

To find the domain of the function \( f(x) = \frac{x^3 - 5x + 3}{x^2 - 1} \), we need to determine where the function is defined. A rational function is undefined when its denominator is equal to zero.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the function is \( x^2 - 1 \).

2. **Set the Denominator Equal to Zero**:
To find the points where the function is undefined, we set the denominator equal to zero:
\[ x^2 - 1 = 0 \]

3. **Solve for \( x \)**:
Factor the equation:
\[ (x - 1)(x + 1) = 0 \]
This gives us the solutions:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

4. **Determine the Points of Undefinedness**:
The function \( f(x) \) is undefined at \( x = 1 \) and \( x = -1 \).

5. **State the Domain**:
The domain of \( f(x) \) includes all real numbers except the points where the function is undefined. Therefore, the domain can be expressed in interval notation as:
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \{-1, 1\} = (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \]

### Final Answer:
The domain of the function \( f(x) = \frac{x^3 - 5x + 3}{x^2 - 1} \) is:
\[ (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \]