The given function is:

\[
f(x) = 3 - 2\sqrt{x - 1}
\]

### Key points:
- This function includes a square root, so we need to ensure the expression under the square root is non-negative to keep the function valid in the real number system.
  
### Domain:
The square root function requires that the argument inside the square root is non-negative, so we must have:

\[
x - 1 \geq 0 \quad \Rightarrow \quad x \geq 1
\]

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

\[
\boxed{x \geq 1}
\]

### Range:
As \( x \geq 1 \), the square root term \( \sqrt{x - 1} \) will take values starting from 0 (when \( x = 1 \)) and increase as \( x \) increases.

- When \( x = 1 \), \( f(1) = 3 - 2\sqrt{1 - 1} = 3 - 2(0) = 3 \).
- As \( x \to \infty \), \( \sqrt{x - 1} \) grows larger, making \( -2\sqrt{x - 1} \) more negative. Hence, \( f(x) \) decreases indefinitely as \( x \to \infty \).

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

\[
\boxed{(-\infty, 3]}
\]

Would you like a more detailed analysis or graph? 

### Related Questions:
1. What is the derivative of \( f(x) = 3 - 2\sqrt{x - 1} \)?
2. How would you compute the inverse of this function?
3. What is the behavior of the function as \( x \to 1^+ \)?
4. Can you find the second derivative of \( f(x) \)?
5. How does the function transform if the square root is replaced by a cube root?

### Tip:
When dealing with square roots in functions, always check the domain carefully since square roots are only defined for non-negative values in the real number system.

Explore how to determine the domain and range of the function f(x) = 3 - 2√(x - 1). This problem focuses on square root functions, domain conditions, and range analysis, with a step-by-step solution. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation