Find the domain and range of the function
`f(x) = sqrt(16 - x^(2))`

Answer

To find the domain and range of the function \( f(x) = \sqrt{16 - x^2} \), we will follow these steps:

### Step 1: Determine the Domain
The expression under the square root must be non-negative for the function to be defined. Therefore, we need to solve the inequality:

\[ 16 - x^2 \geq 0 \]

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

\[ -x^2 \geq -16 \]

Multiplying through by -1 (and reversing the inequality) results in:

\[ x^2 \leq 16 \]

### Step 3: Factor the Inequality
We can factor this as:

\[ x^2 - 16 \leq 0 \]

This can be factored further using the difference of squares:

\[ (x - 4)(x + 4) \leq 0 \]

### Step 4: Find Critical Points
The critical points of the inequality are \( x = -4 \) and \( x = 4 \). We will test intervals around these points to determine where the inequality holds.

### Step 5: Test Intervals
- For \( x < -4 \) (e.g., \( x = -5 \)): \( (-5 - 4)(-5 + 4) = (-9)(-1) > 0 \)
- For \( -4 < x < 4 \) (e.g., \( x = 0 \)): \( (0 - 4)(0 + 4) = (-4)(4) < 0 \)
- For \( x > 4 \) (e.g., \( x = 5 \)): \( (5 - 4)(5 + 4) = (1)(9) > 0 \)

### Step 6: Conclusion for Domain
The inequality \( (x - 4)(x + 4) \leq 0 \) holds true for:

\[ -4 \leq x \leq 4 \]

Thus, the domain of \( f(x) \) is:

\[ \text{Domain: } [-4, 4] \]

### Step 7: Determine the Range
Next, we will find the range of the function \( f(x) = \sqrt{16 - x^2} \).

### Step 8: Analyze the Function
The maximum value of \( f(x) \) occurs when \( x = 0 \):

\[ f(0) = \sqrt{16 - 0^2} = \sqrt{16} = 4 \]

The minimum value occurs when \( x = 4 \) or \( x = -4 \):

\[ f(4) = \sqrt{16 - 4^2} = \sqrt{16 - 16} = 0 \]
\[ f(-4) = \sqrt{16 - (-4)^2} = \sqrt{16 - 16} = 0 \]

### Step 9: Conclusion for Range
Thus, the range of \( f(x) \) is:

\[ \text{Range: } [0, 4] \]

### Final Answer
- **Domain:** \([-4, 4]\)
- **Range:** \([0, 4]\)

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