Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ f(x, y)=y e^{1 / x} $$

Step-by-Step Solution

Answer

The domain of the function \(f(x, y)=ye^{1/x}\) is \((x, y) : x \in \mathbb{R} \backslash {0}, y \in \mathbb{R}\), which signifies that \(x\) can be any real number except zero while \(y\) can be any real number.

Step 1: Determine Domain from the Equation

From the geometric point of view, the domain is the set of \(x\) and \(y\) coordinates in the plane for which the function \(f(x, y)=ye^{1/x}\) is defined. At first glance, the function seems to be defined for all values of \(x\) and \(y\), but there's a factor \(1/x\) up in the exponent of \(e\). The function \(e^{1/x}\) is defined for all \(x\) except at \(x = 0\). For \(y\), there seems to be no restrictions as any real number \(y\) multiplied by an exponential number is still a real number.

Step 2: Formulate the Domain

Therefore, the region \(R\) in the \(xy\)-plane corresponding to the domain of the function \(f(x, y)=ye^{1/x}\) is all real numbers except when \(x = 0\). Formally, this can be expressed as \((x, y) : x \in \mathbb{R} \backslash {0}, y \in \mathbb{R}\).