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Given the function \( h(x, y) = \sqrt{3x - 7y} \), let’s solve each part.

### Part (a): Domain of \( h(x, y) \)

For the function \( h(x, y) \) to be real-valued, the expression inside the square root must be non-negative:

\[
3x - 7y \geq 0
\]

This inequality can be rearranged to:

\[
3x \geq 7y \quad \Rightarrow \quad y \leq \frac{3}{7}x
\]

Thus, the **domain** of \( h \) is all pairs \((x, y)\) where \( y \leq \frac{3}{7}x \).

### Part (b): Range of \( h(x, y) \)

Since \( h(x, y) = \sqrt{3x - 7y} \), the output of \( h \) will always be non-negative because the square root function only produces non-negative values.

1. The minimum value of \( h(x, y) \) is 0, which occurs when \( 3x - 7y = 0 \) (i.e., when \( y = \frac{3}{7}x \)).
2. There is no upper bound for \( h(x, y) \), as \( 3x - 7y \) can grow indefinitely large for points in the domain.

Thus, the **range** of \( h \) is \( [0, \infty) \).

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Would you like further details on any of these steps?

### Related Questions
1. How does the inequality \( y \leq \frac{3}{7}x \) affect the graph of the function \( h(x, y) \)?
2. What changes if the function were \( h(x, y) = \sqrt{7y - 3x} \) instead?
3. How would the domain change if the function was \( h(x, y) = \sqrt{|3x - 7y|} \)?
4. Can you find a pair \((x, y)\) that maximizes \( h(x, y) \)?
5. How does the function behave as \( x \) or \( y \) approach infinity?

### Tip
For functions involving square roots, always check the conditions required to keep the expression inside the root non-negative.

Let h(x, y) = sqrt(3x - 7y).
(a) Find the domain of h.
(b) Find the range of h.

Learn how to determine the domain and range of the function h(x, y) = sqrt(3x - 7y). This problem covers key concepts such as functions of two variables, inequalities, and properties of square root functions, suitable for advanced high school students in Grades 11-12.

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