x = 
1/3

SQRTy
 (y − 3),   9 ≤ y ≤ 16

Given the expression you provided, it appears you're working with a square root function involving a variable \( y \) within a specified range.

The expression seems to be:

\[
x = \frac{1}{3} \sqrt{y - 3}, \quad 9 \leq y \leq 16
\]

Let's break this down:

1. **Expression**: \( x = \frac{1}{3} \sqrt{y - 3} \)
2. **Domain of \( y \)**: \( 9 \leq y \leq 16 \)

### Step-by-Step Breakdown:

1. **Square root function**: The square root part of the function is \( \sqrt{y - 3} \). The value of \( y - 3 \) must be non-negative for the square root to be defined, but since the given domain for \( y \) is \( 9 \leq y \leq 16 \), this is already satisfied.
   
2. **Range of the square root**:
   - When \( y = 9 \), we have \( \sqrt{9 - 3} = \sqrt{6} \).
   - When \( y = 16 \), we have \( \sqrt{16 - 3} = \sqrt{13} \).

3. **Multiply by \( \frac{1}{3} \)**:
   - At \( y = 9 \): \( x = \frac{1}{3} \sqrt{6} \).
   - At \( y = 16 \): \( x = \frac{1}{3} \sqrt{13} \).

### Final Result:

For \( 9 \leq y \leq 16 \), the expression for \( x \) is:

\[
x = \frac{1}{3} \sqrt{y - 3}
\]

If you'd like more details on any specific part of this, let me know!

---

Here are some follow-up questions:

1. How does the domain of \( y \) affect the behavior of \( x \)?
2. What is the maximum value of \( x \) within the given domain?
3. How does the factor \( \frac{1}{3} \) influence the output?
4. What happens to the value of \( x \) as \( y \) increases?
5. Can you express this function in terms of a different variable?

**Tip**: For functions involving square roots, always check the domain to ensure the expression inside the square root is non-negative.

Explore the square root function x = (1/3) * √(y - 3) with the domain 9 ≤ y ≤ 16. This problem involves algebraic concepts, domain and range analysis, and the square root theorem. Step-by-step breakdown included 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