For f(x) = √(x + 5) + 2, what is the domain of f-1?

To determine the domain of \( f^{-1}(x) \), we must first analyze the range of the original function \( f(x) = \sqrt{x + 5} + 2 \). The domain of \( f^{-1}(x) \) is the range of \( f(x) \).

### Step 1: Find the range of \( f(x) \)
The function \( f(x) = \sqrt{x + 5} + 2 \) is defined when:
\[
x + 5 \geq 0 \implies x \geq -5.
\]
Thus, the domain of \( f(x) \) is \( x \geq -5 \).

For \( x \geq -5 \),
\[
f(x) = \sqrt{x + 5} + 2.
\]
The smallest value of \( f(x) \) occurs when \( x = -5 \):
\[
f(-5) = \sqrt{-5 + 5} + 2 = 0 + 2 = 2.
\]
As \( x \to \infty \), \( \sqrt{x + 5} \to \infty \), so \( f(x) \to \infty \).

Thus, the range of \( f(x) \) is:
\[
[2, \infty).
\]

### Step 2: Domain of \( f^{-1}(x) \)
The domain of the inverse function \( f^{-1}(x) \) is the range of \( f(x) \). Therefore:
\[
\text{Domain of } f^{-1}(x) = [2, \infty).
\]

### Final Answer:
The domain of \( f^{-1}(x) \) is:
\[
[2, \infty).
\]

Would you like a detailed explanation of how to compute \( f^{-1}(x) \)? Here are five related questions for exploration:

1. How do you explicitly find \( f^{-1}(x) \) for this function?
2. What is the graph of \( f(x) \) and \( f^{-1}(x) \) together?
3. What transformations occur to the parent function \( \sqrt{x} \) to form \( f(x) \)?
4. How is the domain of \( f^{-1}(x) \) different from the domain of \( f(x) \)?
5. What are practical examples of functions requiring domain and range analysis?

**Tip:** Remember, the domain of \( f^{-1}(x) \) is always equal to

Explore how to determine the domain of the inverse function of f(x) = √(x + 5) + 2. Understand key concepts like the range of the original function, the domain of inverse functions, and square root functions, with a step-by-step explanation. Ideal for high school students in Grades 10-12.

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