Question

Describe the domain and range of the function. $$ f(x, y)=e^{x / y} $$

Short answer

The domain of the function \(f(x, y) = e^{x/y}\) is \((x, y) \in R^2, y ≠ 0\) and its range is \(f(x, y) \in (0, +\infty)\).

Step-by-step solution

Find the Domain

In the function \(f(x, y) = e^{x/y}\), we have 'y' in the denominator of the exponent \(x/y\). Since division by zero is undefined in mathematics, the denominator can't be zero. So, the only restriction for 'x' and 'y' is that 'y' should not be equal to zero. Therefore, the domain of the function is: \((x, y) \in R^2, y ≠ 0\)

Find the Range

This is an exponential function and the base of the exponential function 'e' is a constant. The output of an exponential function is always positive irrespective of the input. Here, for any real numbers 'x' and 'y' (where \(y ≠ 0\)), \(e^{x/y}\) is always going to be positive and not zero, since any number raised to a power is always greater than zero. Therefore, the range of the function is: \(f(x, y) \in (0, +\infty)\)