Find the domain and range of the following functions:
`sqrt(1-x^(2))`

Answer

To find the domain and range of the function \( f(x) = \sqrt{1 - x^2} \), we will follow these steps:

### Step 1: Determine the Domain
The function \( f(x) = \sqrt{1 - x^2} \) is defined only when the expression inside the square root is non-negative. Therefore, we need to solve the inequality:

\[ 1 - x^2 \geq 0 \]

### Step 2: Rearranging the Inequality
Rearranging the inequality gives us:

\[ 1 \geq x^2 \]

This can also be written as:

\[ x^2 \leq 1 \]

### Step 3: Finding the Critical Points
To solve \( x^2 \leq 1 \), we can find the critical points by setting \( x^2 = 1 \):

\[ x = -1 \quad \text{and} \quad x = 1 \]

### Step 4: Testing Intervals
Next, we test the intervals determined by these critical points: \( (-\infty, -1) \), \( (-1, 1) \), and \( (1, \infty) \).

- For \( x < -1 \) (e.g., \( x = -2 \)):
\[ (-2)^2 = 4 \quad \Rightarrow \quad 4 > 1 \quad \text{(not in the domain)} \]

- For \( -1 < x < 1 \) (e.g., \( x = 0 \)):
\[ 0^2 = 0 \quad \Rightarrow \quad 0 \leq 1 \quad \text{(in the domain)} \]

- For \( x > 1 \) (e.g., \( x = 2 \)):
\[ 2^2 = 4 \quad \Rightarrow \quad 4 > 1 \quad \text{(not in the domain)} \]

### Step 5: Conclusion on Domain
Thus, the function is defined for \( x \) in the interval:

\[ [-1, 1] \]

### Step 6: Determine the Range
Now, we will determine the range of the function \( f(x) = \sqrt{1 - x^2} \).

### Step 7: Finding the Maximum and Minimum Values
The expression \( 1 - x^2 \) reaches its maximum value when \( x = 0 \):

\[ f(0) = \sqrt{1 - 0^2} = \sqrt{1} = 1 \]

The minimum value occurs when \( x = -1 \) or \( x = 1 \):

\[ f(-1) = \sqrt{1 - (-1)^2} = \sqrt{0} = 0 \]
\[ f(1) = \sqrt{1 - 1^2} = \sqrt{0} = 0 \]

### Step 8: Conclusion on Range
Thus, the range of \( f(x) \) is:

\[ [0, 1] \]

### Final Answer
- **Domain**: \([-1, 1]\)
- **Range**: \([0, 1]\)