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Chapter 12: Problem 20

Find the domain of the following functions. $$h(x, y)=\sqrt{x-2 y+4}$$

Short Answer

Expert verified
Question: Determine the domain of the function $$h(x, y) = \sqrt{x-2y+4}$$. Answer: The domain of the function is given by the set of all points $$(x, y)$$ that satisfy the inequality $$x-2y \geq -4$$. In terms of the restrictions for $$x$$ and $$y$$, the domain can be expressed as: $$\{(x, y) \mid x \geq -4, y \leq 2\}$$.

Step by step solution

01

Solve the inequality

First, we need to solve the inequality for the function: $$x-2 y+4 \geq 0$$ To solve this inequality, let's rewrite it as: $$x - 2y \geq -4$$
02

Find the domain

Now, we need to find the set of all points $$(x, y)$$ for which the inequality $$x - 2y \geq -4$$ holds true. 1) If $$x=0$$, $$0 - 2y \geq -4$$ $$-2y \geq -4$$ $$y \leq 2$$ 2) If $$y=0$$, $$x - 2(0) \geq -4$$ $$x \geq -4$$ Putting the results from both conditions together, we find that to satisfy the inequality, $$x \geq -4$$ and $$y \leq 2$$. This means that any point $$(x, y)$$ that falls within these restrictions will result in a real output for the function $$h(x, y)$$.
03

Express the domain

We can express the domain as a set of points $$(x, y)$$ that satisfy the inequality $$x - 2y \geq -4$$. In terms of the restrictions we found for $$x$$ and $$y$$, the domain can be written as: $$\{(x, y) \mid x \geq -4, y \leq 2\}$$ This is the domain of the function $$h(x, y) = \sqrt{x - 2y + 4}$$.

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