Question

Describe the domain and range of the function. $$ f(x, y)=\arccos (y / x) $$

Short answer

The domain of the function are all pairs (x, y) in the real numbers, such that \(-x \leq y \leq x\) for \(x > 0\) and \(x \leq y \leq -x\) for \(x < 0\). Pairs of the form (0, y) are excluded. The range of the function is between 0 and \(\pi\).

Step-by-step solution

Define the domain

First, recognize that for the function \(f(x, y)=\arccos (y / x)\), \(y/x\) must be between -1 and 1 inclusive for the function to be defined. And also \(x\neq 0\). Therefore, the domain of the function are all pairs (x, y) in the real numbers, such that \(-x \leq y \leq x\) for \(x > 0\) and \(x \leq y \leq -x\) for \(x < 0\). No restrictions on y when x=0, hence pairs of the form (0, y) for any y are excluded.

Define the range

Since this is an arccosine function, we also know that the range is between 0 and \(\pi\). The arccosine function only returns these values.