To find the **domain** and **range** of the function \( f(x) = \sqrt{x+16} \), let's break it down:

### Domain:
The expression inside the square root, \( x+16 \), must be **non-negative** because the square root of a negative number is not defined in the set of real numbers.

\[
x + 16 \geq 0
\]

Solving this inequality:

\[
x \geq -16
\]

So, the domain of the function is all values of \( x \) that are greater than or equal to \(-16\). In interval notation, the domain is:

\[
\text{Domain} = [-16, \infty)
\]

### Range:
For the range, consider the output of the function \( f(x) = \sqrt{x+16} \). Since the square root function always gives non-negative values, the smallest value of \( \sqrt{x+16} \) occurs when \( x = -16 \), and that value is:

\[
f(-16) = \sqrt{-16 + 16} = \sqrt{0} = 0
\]

As \( x \) increases, the value of \( \sqrt{x+16} \) increases without bound. Therefore, the range of the function is:

\[
\text{Range} = [0, \infty)
\]

### Summary:
- **Domain**: \( [-16, \infty) \)
- **Range**: \( [0, \infty) \)

Would you like more details or have any further questions?

Here are 5 related questions you could explore:
1. What is the domain and range of \( \sqrt{x-5} \)?
2. How would the domain and range change if the function were \( f(x) = \sqrt{-x+16} \)?
3. What is the domain and range of \( \sqrt{x^2 + 4x + 4} \)?
4. How would you find the domain and range of a cubic function like \( f(x) = x^3 + 2x^2 + 3x + 1 \)?
5. How does adding a constant outside the square root, like \( f(x) = \sqrt{x+16} + 2 \), affect the range?

**Tip:** Always check for constraints like non-negative values inside square roots and undefined points (like division by zero) when determining the domain of a function.

Discover how to find the domain and range of the function f(x) = √(x+16). This problem involves key concepts in algebra, focusing on square roots and function analysis. Suitable for high school students in Grades 9-11, the solution explores restrictions on the input and output values of the function.

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