I was given the following function: $$f(x, y) := \begin{cases} 2 - \frac{1}{x^2\,+\,y^2} & \text{if } 0<x^2 + y^2 \leqslant 1 \\ \sqrt{x^2 + y^2-1} & \text{if } 1 < x^2 + y^2 \end{cases} \quad$$ How can I find it's range? I'm conviced it is the interval $(0, 1]$, by simple calculation. Is it right? It's simple, but I'm a little concerned about the disjoint intervals of the ranges of the piecewise functions. Thanks in advance!
2026-04-06 04:18:00.1775449080
Range of a function of two variables given by two sentences
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Hint:
I'm not sure why you consider the range is $(0,1]$. Instead, for the first part, consider what happens to $f(x,y) = 2 - \frac{1}{x^2 + y^2}$ as $x^2 + y^2 \to 0^{+}$. For the second part, consider what happens to $f(x,y) = \sqrt{x^2 + y^2 - 1}$ as $x^2 + y^2 \to \infty$.