Question

Find the inverse of the function x2 + y2 = 4 and the domain of the inverse for 0 ≤ x ≤ 2.

Answer 1

To get the inverse "relation" of an expression, we first off, do a quick switcharoo of the variables, and then solve for "y", so let's proceed,

$\bf x^2+y^2=4\qquad inverse\implies \boxed{y}^2+\boxed{x}^2=4 \\\\\\ y^2=4-x^2\implies y=\pm\sqrt{4-x^2}$

and yes, the domain for the range 0 ⩽ x ⩽ 2, let's get instead the "range" of the original function,

$\bf x^2+y^2=4\implies 0^2+y^2=4\implies y=\pm\sqrt{4}\implies \boxed{y=\pm 2} \\\\\\ x^2+y^2=4\implies 2^2+y^2=4\implies 4+y^2=4\implies \boxed{y=0}\\\\ -------------------------------\\\\ \stackrel{\textit{range of original}}{-2\le y \le 2}~~=~~\stackrel{\textit{domain of its inverse}}{-2\le x \le 2}$