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Chapter 11: Problem 18

Describe the domain and range of the function. $$ f(x, y)=\sqrt{4-x^{2}-4 y^{2}} $$

Short Answer

Expert verified
The domain of the function \(f(x, y) = \sqrt{4 - x^{2} - 4y^{2}}\) is all ordered pairs \((x, y)\) such that \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\) or \(x^{2} + y^{2} \leq 1\). The range of the function is \([0,2]\).

Step by step solution

01

Define the function

The function is defined as \(f(x, y) = \sqrt{4 - x^{2} - 4y^{2}}\)
02

Find the domain

The expression under the square root must be zero or positive. This gives us the inequality \(4 - x^{2} - 4y^{2} \geq 0\). This simplifies to \(x^{2} + y^{2} \leq 1\), which is the equation of a circle with radius 1. Therefore, the domain is all ordered pairs \((x, y)\) such that \(x^{2} + y^{2} \leq 1\) or \((-1 \leq x \leq 1\) and \(-1 \leq y \leq 1)\).
03

Find the range

Since the square root function doesn鈥檛 yield negative values, the range of \(f(x, y)\) should be \([0, +\infty)\). However, considering the restriction of the domain, the maximum value of the function is indeed 2 (which happens when x=y=0). This gives us the range \([0,2]\).

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