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 the values of \( x \) for which the function is defined. The function will be undefined where the denominator equals zero.

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

### Step 2: Set the denominator equal to zero
To find the values that make the function undefined, we set the denominator equal to zero:
\[ x^2 - 1 = 0 \]

### Step 3: Solve for \( x \)
Now, we solve the equation:
\[ x^2 = 1 \]
Taking the square root of both sides, we get:
\[ x = 1 \quad \text{or} \quad x = -1 \]

### Step 4: Determine the domain
The function \( f(x) \) is undefined at \( x = 1 \) and \( x = -1 \). Therefore, the domain of \( f(x) \) includes all real numbers except these two points.

Thus, we can express the domain in interval notation:
\[ \text{Domain of } f(x) = (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \]

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

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