The function \( f(x) = \sqrt{x} - 1 \) represents a transformation of the basic square root function \( \sqrt{x} \).

### Key Properties:
1. **Domain:** The function \( \sqrt{x} \) is only defined for \( x \geq 0 \). Therefore, the domain of \( f(x) \) is all \( x \) values for which \( x \geq 0 \).
2. **Range:** The square root function \( \sqrt{x} \) has a range of \( [0, \infty) \). Subtracting 1 shifts the range down by 1, so the range of \( f(x) \) is \( [-1, \infty) \).
3. **Intercepts:**
   - **x-intercept:** Set \( f(x) = 0 \) and solve for \( x \):
     \[
     \sqrt{x} - 1 = 0 \implies \sqrt{x} = 1 \implies x = 1
     \]
     So, the x-intercept is at \( (1, 0) \).
   - **y-intercept:** Set \( x = 0 \):
     \[
     f(0) = \sqrt{0} - 1 = -1
     \]
     So, the y-intercept is at \( (0, -1) \).

4. **Behavior:**
   - As \( x \) increases, \( f(x) \) increases.
   - The function is continuous and smooth for all \( x \geq 0 \).

Would you like further details or have any questions?

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**Related Questions:**
1. How would you graph the function \( f(x) = \sqrt{x} - 1 \)?
2. What are the transformations involved in the function \( f(x) = \sqrt{x} - 1 \)?
3. How does the domain of \( f(x) = \sqrt{x} - 1 \) differ from \( f(x) = \sqrt{x} + 1 \)?
4. Can you find the derivative of \( f(x) = \sqrt{x} - 1 \)?
5. What is the second derivative of \( f(x) = \sqrt{x} - 1 \) and what does it represent?
6. How would you find the inverse of the function \( f(x) = \sqrt{x} - 1 \)?
7. What happens to the function \( f(x) = \sqrt{x} - 1 \) as \( x \) approaches infinity?
8. How do the intercepts of \( f(x) = \sqrt{x} - 1 \) change if the function is modified to \( f(x) = \sqrt{x - 1} \)?

**Tip:** When analyzing functions, always start by determining the domain and range, as they set the foundation for understanding the function's behavior.

Explore the function f(x) = sqrt(x) - 1, its domain, range, intercepts, and behavior. This detailed explanation covers key mathematical concepts related to functions and transformations, suitable for students in Grades 10-12.

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