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Chapter 7: Problem 2

For each function, find the domain. $$f(x, y)=\frac{\sqrt{x}}{\sqrt{y}}$$

Short Answer

Expert verified
The domain is \( x \geq 0 \) and \( y > 0 \).

Step by step solution

01

Identify Restrictions from the Square Roots

For the function \( f(x, y) = \frac{\sqrt{x}}{\sqrt{y}} \), both the numerator and the denominator involve square roots. The square root function is defined only for non-negative inputs. Therefore, we need \( x \geq 0 \) and \( y \geq 0 \).
02

Identify Restrictions from the Denominator

The denominator of the function is \( \sqrt{y} \). Since division by zero is undefined, \( \sqrt{y} \) must be greater than zero, which implies \( y > 0 \).
03

Combine the Conditions

To find the domain, combine the conditions from Steps 1 and 2. We have \( x \geq 0 \) and \( y > 0 \). Therefore, the domain of the function is all \( (x, y) \) where \( x \geq 0 \) and \( y > 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Restrictions
In mathematics, square root restrictions are essential to determine the domain of a function. When dealing with square roots, it鈥檚 crucial to remember that the square root of a negative number is not defined within the realm of real numbers. Therefore, to ensure the function operates within real numbers, the expression under the square root must be non-negative.

For the function \( f(x, y) = \frac{\sqrt{x}}{\sqrt{y}} \), this restriction tells us that:
  • \( x \geq 0 \) because the square root function \( \sqrt{x} \) needs a non-negative number to provide a real output.
  • \( y \geq 0 \) for the square root in the denominator function \( \sqrt{y} \), as well.
This step ensures we are working with numbers within the permissible range, granting the function validity in real number terms.
Denominator Restrictions
Another vital concept when finding the domain is addressing denominator restrictions. Specifically, algebra dictates that a function cannot have a zero denominator, as division by zero is undefined.

In the given function \( f(x, y) = \frac{\sqrt{x}}{\sqrt{y}} \), the denominator \( \sqrt{y} \) should not be zero. Though previously we mentioned that \( y \geq 0 \) for the square root, the added necessity here for the denominator means \( y \) must be strictly greater than zero:
  • Hence, \( y > 0 \) to avoid dividing by zero.
This condition is paramount to maintain the structural integrity of the function as mathematical operations involved must be valid.
Inequality Constraints
Understanding inequality constraints allows us to combine various conditions that define the function's domain comprehensively.

By integrating square root and denominator restrictions, we have:
  • \( x \geq 0 \) from square root considerations.
  • \( y > 0 \) owing to both square root and denominator restrictions.
The effective domain emerges by these combined conditions, meaning the possible inputs (\( x, y \)) must satisfy both of these inequalities.

Thus, only pairs (\( x, y \)) where \( x \geq 0 \) and \( y > 0 \) allow the function \( f(x, y) \) to yield a real and defined output, characterizing its domain successfully. This systematic approach ensures a complete understanding of the interplay between different constraints in determining the function's validity.

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Most popular questions from this chapter

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