Find domain of `f(x)=log_(10)(1+x^(3))`.

Answer

To find the domain of the function \( f(x) = \log_{10}(1 + x^3) \), we need to ensure that the argument of the logarithm is positive. The logarithm function is defined only for positive values. Therefore, we need to solve the inequality:

\[ 1 + x^3 > 0 \]

### Step 1: Solve the inequality
We can rearrange the inequality:

\[ x^3 > -1 \]

### Step 2: Find the critical point
To find the critical point where the inequality changes, we can set \( x^3 = -1 \):

\[ x = -1 \]

### Step 3: Test intervals
Now we will test the intervals around the critical point \( x = -1 \):

1. **Interval 1**: \( x < -1 \) (e.g., \( x = -2 \))
- \( (-2)^3 = -8 \) → \( 1 + (-8) = -7 \) (not valid)

2. **Interval 2**: \( x = -1 \)
- \( (-1)^3 = -1 \) → \( 1 + (-1) = 0 \) (not valid)

3. **Interval 3**: \( x > -1 \) (e.g., \( x = 0 \))
- \( (0)^3 = 0 \) → \( 1 + 0 = 1 \) (valid)

### Step 4: Conclusion
From the testing, we find that the inequality \( 1 + x^3 > 0 \) holds true for all \( x > -1 \). Thus, the domain of the function is:

\[ \text{Domain} = (-1, \infty) \]

### Final Answer
The domain of \( f(x) = \log_{10}(1 + x^3) \) is \( x \in (-1, \infty) \).

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