Question

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

Short 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-by-step solution

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.

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 \((x, y) : x \in \mathbb{R} \backslash {0}, y \in \mathbb{R}\).