Homework: MLM 15.1 craphs Question 4, 15.1.21 Points: 0 of 1 Save and Level Curves Find the domain of the following function. estion list Question 1 Question 2 Question 3 Question 4 \[ g(x, y)=\ln \left(x^{5}-y\right) \] Select the correct choice below and fill in any answer boxes in your choice. A. \( \{(x, y): x \neq \) \( \square \) and \( y \) z \( \square \) ) (Use a comma to separate answers as needed.) B. \( \{(x, y): y<\square \) \( \square \) C. \( \{(x, y): y\rangle \square \) D. \( \{(x, y): y \neq \square\} \) \( \square \) E. \( R^{2} \)

To find the domain of the function \( g(x, y) = \ln(x^5 - y) \), we need to ensure the argument of the logarithm is positive. This means we require \( x^5 - y > 0 \), which simplifies to \( y < x^5 \). So the domain includes all points \((x, y)\) where \(y\) is less than \(x^5\). Now, to imagine this visually, the curve \(y = x^5\) acts as a boundary. Everything below this curve (where \(y\) takes smaller values than \(x^5\)) is included in the domain. Just visualize it: as \(x\) increases, \(x^5\) rises quickly, creating a sweeping curve in the 2D plane where all points beneath it are fair game! For further exploration, if you’re interested in the characteristics of logarithmic functions, a fun fact is that they are defined for positive arguments only! This means the function will show interesting behaviors like vertical asymptotes and rapid increases, adding layers to their graph. Diving into applications of these properties in fields like engineering and economics can show how crucial understanding domains is!