What is the domain and range of the function f(x ) = ( x + 13)^1/2  - 3   ?

To determine the domain and range of the function \( f(x) = \sqrt{x + 13} - 3 \), let's analyze it step by step.

### 1. Domain:
The domain of a function is the set of all possible input values (i.e., \( x \)) for which the function is defined.

- The function involves a square root, \( \sqrt{x + 13} \). For the square root to be defined, the expression under the square root must be non-negative.
  
  \[
  x + 13 \geq 0
  \]
  \[
  x \geq -13
  \]
  
- Therefore, the domain of the function is all \( x \) such that \( x \geq -13 \).

So, the domain is:
\[
\text{Domain: } [-13, \infty)
\]

### 2. Range:
The range of a function is the set of all possible output values (i.e., \( f(x) \)).

- The smallest value inside the square root occurs when \( x = -13 \), which gives \( \sqrt{0} = 0 \).
  
- Substituting \( x = -13 \) into the function:
  \[
  f(-13) = \sqrt{-13 + 13} - 3 = \sqrt{0} - 3 = -3
  \]
  
- As \( x \) increases beyond \(-13\), \( \sqrt{x + 13} \) increases, so \( f(x) = \sqrt{x + 13} - 3 \) also increases.

- Since \( \sqrt{x + 13} \) can be arbitrarily large as \( x \) increases, \( f(x) \) can also be arbitrarily large.

Therefore, the range of the function is:
\[
\text{Range: } [-3, \infty)
\]

### Summary:
- **Domain:** \( [-13, \infty) \)
- **Range:** \( [-3, \infty) \)

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. How does shifting a square root function horizontally affect its domain?
2. What is the impact of vertical shifts on the range of a function?
3. Can a square root function ever have a negative output?
4. How do you find the domain and range for a function involving absolute values?
5. What are the transformations of the parent function \( \sqrt{x} \)?

**Tip:** When finding the domain of a function involving a square root, always check the expression inside the square root to ensure it's non-negative.

Learn how to determine the domain and range of the function f(x) = sqrt(x + 13) - 3. Understand the conditions for the square root function to be defined and how it affects the domain. Explore how the output values vary based on different input values. Suitable for high school students.

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