Find the range of the function. (Enter your answer using interval notation.)
V(x, y) = 6x2 + y2
 

We are tasked with finding the range of the function:

\[
V(x, y) = 6x^2 + y^2
\]

### Step 1: Analyze the function structure
The given function is a sum of two non-negative terms:
- \( 6x^2 \) is non-negative because \( x^2 \geq 0 \) for all real \( x \), and multiplying by 6 still keeps it non-negative.
- \( y^2 \) is also non-negative since \( y^2 \geq 0 \) for all real \( y \).

The minimum value of \( 6x^2 + y^2 \) occurs when both \( x = 0 \) and \( y = 0 \), which gives:

\[
V(0, 0) = 6(0)^2 + (0)^2 = 0
\]

### Step 2: Determine the maximum value
Since both \( 6x^2 \) and \( y^2 \) are non-negative and unbounded as \( |x| \) or \( |y| \) increase, the function \( V(x, y) \) has no upper bound. As \( x \) or \( y \) approach infinity, the value of the function also tends to infinity.

### Step 3: Conclusion
The range of the function is all real values starting from 0 and extending to infinity. Therefore, the range is:

\[
\boxed{[0, \infty)}
\]

Would you like more details on this or have any questions?

Here are 5 related questions:
1. What is the domain of the function \( V(x, y) = 6x^2 + y^2 \)?
2. How would the range change if the function was \( V(x, y) = -6x^2 - y^2 \)?
3. What are the level curves of the function \( V(x, y) \)?
4. How does the range of a quadratic function of two variables generally behave?
5. What happens to the range if we modify \( V(x, y) \) to \( 6x^2 + y^4 \)?

**Tip**: The range of a function is closely related to the type of terms involved (e.g., squares are always non-negative). Look at each term to determine how it affects the output.

Find the range of the function. (Enter your answer using interval notation.) V(x, y) = 6x^2 + y^2

Discover how to find the range of the function V(x, y) = 6x^2 + y^2 using interval notation. This problem explores quadratic functions and their properties. Suitable for students in Grades 10-12, the solution includes key concepts in multivariable calculus and algebra.

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