Find the domain the range of each of the following function: `f(x)=sin^(2)(x^(3))+cos^(2)(x^(3))`

Explanation

To find the domain and range of the function \( f(x) = \sin^2(x^3) + \cos^2(x^3) \), we can follow these steps:

### Step 1: Identify the function
We start with the function:
\[ f(x) = \sin^2(x^3) + \cos^2(x^3) \]

### Step 2: Use the Pythagorean identity
Recall the Pythagorean identity:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
for any angle \(\theta\). In our case, we can let \(\theta = x^3\). Therefore, we can rewrite the function as:
\[ f(x) = 1 \]

### Step 3: Determine the domain
The function \(f(x) = 1\) is defined for all values of \(x\) since there are no restrictions on the sine and cosine functions. Thus, the domain of \(f(x)\) is:
\[ \text{Domain} = \mathbb{R} \quad \text{(all real numbers)} \]

### Step 4: Determine the range
Since \(f(x) = 1\) for all \(x\), the output of the function does not vary; it is always 1. Therefore, the range of \(f(x)\) is:
\[ \text{Range} = \{1\} \]

### Final Answer
- **Domain**: \(\mathbb{R}\) (all real numbers)
- **Range**: \(\{1\}\)