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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 \( \{(x, y) \mid x \geq 0, y > 0\} \).

Step by step solution

01

Identify Restrictions for Square Roots

For the function \( f(x, y) = \frac{\sqrt{x}}{\sqrt{y}} \), notice that both \( \sqrt{x} \) and \( \sqrt{y} \) have restrictions. The square root function is only defined for non-negative numbers, so \( x \geq 0 \) and \( y > 0 \) since it is in the denominator.
02

Ensure Non-Zero Denominator

To ensure the denominator \( \sqrt{y} eq 0 \), \( y \) must be strictly positive. Therefore, \( y > 0 \) is a condition that must be satisfied to avoid division by zero.
03

Combine Conditions

Combine the conditions from previous steps: \( x \) must be greater than or equal to zero, and \( y \) must be greater than zero. Thus, the domain of the function \( f(x, y) \) is all points \( (x, y) \) 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.

Restrictions of Square Root Functions
Square root functions have certain restrictions that define where they can be used. The main restriction is that the number under the square root, known as the radicand, must be non-negative. In mathematical terms, this means the expression inside the square root must be greater than or equal to zero.
For example, to find the domain of a function like \( \sqrt{x} \), you must ensure \( x \geq 0 \).
In the case of the given function \( f(x, y) = \frac{\sqrt{x}}{\sqrt{y}} \), this means:
  • \( x \) must be at least zero, or \( x \geq 0 \), because \( \sqrt{x} \) must have a non-negative input.
  • \( y \) must be positive, or \( y > 0 \), because it appears in the denominator, and also because the square root function is involved, making \( \sqrt{y} \) non-negative with \( y eq 0 \).
Denominator Cannot Be Zero
When you have a fraction, the denominator鈥攖he bottom part of the fraction鈥攈as to be non-zero to avoid undefined situations in mathematics.
In simple terms, if you divide by zero, you end up with an undefined result, which does not work in math operations.
For the function \( f(x, y) = \frac{\sqrt{x}}{\sqrt{y}} \), we focus especially on \( \sqrt{y} \) in the denominator. Here, it is crucial that \( y > 0 \) not just \( y \geq 0 \). This ensures that \( \sqrt{y} eq 0 \), complying with the requirement to keep the denominator from equating to zero.
By ensuring \( y > 0 \), we confirm that the function remains valid and defined.
Combining Conditions for Domain Determination
To figure out the overall domain of a function, it is often necessary to consider multiple conditions together.
This function, \( f(x, y) = \frac{\sqrt{x}}{\sqrt{y}} \), serves as a classic example of this process.
We've already identified the individual restrictions:
  • The square root restrictions required \( x \geq 0 \) and, due to inclusion in the fraction, \( y > 0 \).
  • The denominator requirement further gives the restriction \( y > 0 \).
By combining these conditions, we see the domain of the function is all pairs \((x, y)\) such that \( x \geq 0 \) and \( y > 0 \).
This means you can input any non-negative \( x \) and a positive \( y \) into the function, ensuring the calculations remain valid and obey all mathematical rules.

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