Find the domain of the function : `f(x)=3/(4-x^2)+(log)_(10)(x^3-x)`

Explanation

To find the domain of the function \( f(x) = \frac{3}{4 - x^2} + \log_{10}(x^3 - x) \), we need to determine the values of \( x \) for which this function is defined. This requires us to consider both components of the function: the rational part and the logarithmic part.

### Step 1: Analyze the rational part \( \frac{3}{4 - x^2} \)

The rational function is defined as long as the denominator is not zero. Therefore, we need to solve the inequality:

\[ 4 - x^2 \neq 0 \]

This simplifies to:

\[ x^2 \neq 4 \]

From this, we find:

\[ x \neq 2 \quad \text{and} \quad x \neq -2 \]

### Step 2: Analyze the logarithmic part \( \log_{10}(x^3 - x) \)

The logarithmic function is defined only for positive arguments. Thus, we need to solve the inequality:

\[ x^3 - x > 0 \]

Factoring gives us:

\[ x(x^2 - 1) > 0 \]

This can be factored further:

\[ x(x - 1)(x + 1) > 0 \]

### Step 3: Determine the critical points

The critical points from the factors are:

- \( x = 0 \)
- \( x = 1 \)
- \( x = -1 \)

### Step 4: Test intervals on a number line

We will test the sign of \( x(x - 1)(x + 1) \) in the intervals determined by the critical points: \( (-\infty, -1) \), \( (-1, 0) \), \( (0, 1) \), and \( (1, \infty) \).

1. **Interval \( (-\infty, -1) \)**: Choose \( x = -2 \)
- \( (-2)(-2 - 1)(-2 + 1) = (-2)(-3)(-1) < 0 \)

2. **Interval \( (-1, 0) \)**: Choose \( x = -0.5 \)
- \( (-0.5)(-0.5 - 1)(-0.5 + 1) = (-0.5)(-1.5)(0.5) > 0 \)

3. **Interval \( (0, 1) \)**: Choose \( x = 0.5 \)
- \( (0.5)(0.5 - 1)(0.5 + 1) = (0.5)(-0.5)(1.5) < 0 \)

4. **Interval \( (1, \infty) \)**: Choose \( x = 2 \)
- \( (2)(2 - 1)(2 + 1) = (2)(1)(3) > 0 \)

### Step 5: Combine results

From the sign analysis, we find that \( x(x - 1)(x + 1) > 0 \) in the intervals:

- \( (-1, 0) \)
- \( (1, \infty) \)

### Step 6: Exclude points where the rational part is undefined

We must also exclude \( x = 2 \) and \( x = -2 \) from our domain.

### Final Domain

Combining the results from the rational and logarithmic parts, we find:

\[ \text{Domain of } f(x) = (-1, 0) \cup (1, 2) \cup (2, \infty) \]