https://file.aiquickdraw.com/mathbot/file/1725933814606cpecgbdh.jpg

Let's analyze each equation to determine if it defines \( y \) as a function of \( x \). A function means that for each value of \( x \), there is exactly one corresponding value of \( y \).

1. \( x^2 + 4y = 6 \)
   - Solving for \( y \), we get \( y = \frac{6 - x^2}{4} \). This equation defines \( y \) uniquely in terms of \( x \).  
   **Answer: Yes**

2. \( x^2y + y = 6 \)
   - Factoring the \( y \), we get \( y(x^2 + 1) = 6 \), or \( y = \frac{6}{x^2 + 1} \). Since this gives a unique value of \( y \) for each \( x \), this is a function.  
   **Answer: Yes**

3. \( 2|x| + y = 10 \)
   - Solving for \( y \), we get \( y = 10 - 2|x| \). The absolute value introduces no ambiguity, as for each \( x \), there is still only one value of \( y \).  
   **Answer: Yes**

4. \( 9 + x = y^3 \)
   - Solving for \( y \), we get \( y = \sqrt[3]{9 + x} \). The cube root function is well-defined and provides a unique value for \( y \) for each \( x \).  
   **Answer: Yes**

In summary, all four equations define \( y \) as a function of \( x \).

Would you like further details on any of the steps? Here are 5 related questions to deepen your understanding:

1. How does solving for \( y \) determine whether an equation defines a function?
2. What distinguishes a function from a relation in mathematics?
3. Can an equation involving absolute values still define a function?
4. How can you determine if a multivariable equation defines one variable as a function of the other?
5. What would happen if we had a square root instead of a cube root in equation 4?

**Tip:** When verifying if an equation defines a function, always check whether solving for \( y \) results in a unique value for every \( x \).

Enter Yes or No in each answer space below to indicate whether the corresponding equation defines y as a function of x: 
1. x^2 + 4y = 6 
2. x^2y + y = 6 
3. 2|x| + y = 10 
4. 9 + x = y^3

Analyze equations such as x^2 + 4y = 6 and 9 + x = y^3 to determine if they define y as a function of x. This involves key concepts like algebraic equations, absolute values, and cube roots. Ideal for high school students (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