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Find the domain of f(x, y) = √√x - 4 + ln(y³ + x).

Find the domain of f(x, y) = √√x - 4 + ln(y³ + x).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Find the domain of \( f(x, y) = \sqrt{x - 4} + \ln(y^3 + x) \). **Explanation:** The function \( f(x, y) \) includes two components that dictate its domain: 1. **Square Root Component \( \sqrt{x - 4} \):** - The expression inside the square root, \( x - 4 \), must be greater than or equal to zero for the square root to be defined in the real number system. - Therefore, \( x - 4 \geq 0 \) which simplifies to \( x \geq 4 \). 2. **Logarithmic Component \( \ln(y^3 + x) \):** - The argument of the natural logarithm function, \( y^3 + x \), must be greater than zero for the logarithm to be defined. - This implies \( y^3 + x > 0 \). **Domain:** The domain of \( f(x, y) \) is the set of all pairs \((x, y)\) that satisfy both conditions simultaneously: - \( x \geq 4 \) - \( y^3 + x > 0 \) In summary, identify the values of \( x \) and \( y \) that will satisfy both conditions for the function to be defined.
Transcribed Image Text:**Problem Statement:** Find the domain of \( f(x, y) = \sqrt{x - 4} + \ln(y^3 + x) \). **Explanation:** The function \( f(x, y) \) includes two components that dictate its domain: 1. **Square Root Component \( \sqrt{x - 4} \):** - The expression inside the square root, \( x - 4 \), must be greater than or equal to zero for the square root to be defined in the real number system. - Therefore, \( x - 4 \geq 0 \) which simplifies to \( x \geq 4 \). 2. **Logarithmic Component \( \ln(y^3 + x) \):** - The argument of the natural logarithm function, \( y^3 + x \), must be greater than zero for the logarithm to be defined. - This implies \( y^3 + x > 0 \). **Domain:** The domain of \( f(x, y) \) is the set of all pairs \((x, y)\) that satisfy both conditions simultaneously: - \( x \geq 4 \) - \( y^3 + x > 0 \) In summary, identify the values of \( x \) and \( y \) that will satisfy both conditions for the function to be defined.
Expert Solution
Step 1

Note-1 We should know that root function has always domain positive real numbers.

Note-2 ln function have positive real numbers as domain.

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