To solve the expression \( \sqrt{x-1} + \sqrt{4-x} \), let's first understand the domain of the square roots to ensure the expression is valid.

### Step 1: Define the domain
For both square roots to be defined (non-negative values under the square root), the following conditions must hold:

1. \( x - 1 \geq 0 \Rightarrow x \geq 1 \)
2. \( 4 - x \geq 0 \Rightarrow x \leq 4 \)

Thus, the valid domain for \( x \) is:
\[
1 \leq x \leq 4
\]

### Step 2: Simplify the expression for specific values of \( x \)

To get a clearer idea of how the expression behaves, let's evaluate the expression at some values within the domain.

- At \( x = 1 \):
  \[
  \sqrt{1 - 1} + \sqrt{4 - 1} = \sqrt{0} + \sqrt{3} = 0 + \sqrt{3} \approx 1.732
  \]

- At \( x = 4 \):
  \[
  \sqrt{4 - 1} + \sqrt{4 - 4} = \sqrt{3} + \sqrt{0} = \sqrt{3} + 0 \approx 1.732
  \]

- At \( x = 2 \):
  \[
  \sqrt{2 - 1} + \sqrt{4 - 2} = \sqrt{1} + \sqrt{2} = 1 + \sqrt{2} \approx 1 + 1.414 = 2.414
  \]

- At \( x = 3 \):
  \[
  \sqrt{3 - 1} + \sqrt{4 - 3} = \sqrt{2} + \sqrt{1} = \sqrt{2} + 1 \approx 1.414 + 1 = 2.414
  \]

### Conclusion:
The expression \( \sqrt{x-1} + \sqrt{4-x} \) is valid in the domain \( 1 \leq x \leq 4 \) and takes values depending on \( x \). The exact form of the expression cannot be simplified further without specifying \( x \).

Would you like me to further explore this or graph the function?

---

Here are 5 related questions to deepen your understanding:

1. How do you determine the domain of a function with square roots?
2. What happens to the value of the expression as \( x \) approaches the boundaries of the domain?
3. Can you solve the equation \( \sqrt{x-1} + \sqrt{4-x} = 3 \)?
4. How does the concept of continuity apply to the function \( \sqrt{x-1} + \sqrt{4-x} \)?
5. What is the significance of defining the domain before solving expressions?

**Tip:** Always check the domain for square root functions to avoid dealing with undefined expressions!

Explore how to find the domain of the expression √(x-1) + √(4-x) and evaluate it at specific values of x. Learn key algebraic concepts such as square roots and the conditions for their domain, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation