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Chapter 11: Problem 24

Describe the domain and range of the function. $$ z=\frac{x y}{x-y} $$

Short Answer

Expert verified
The domain of the function \(z=\frac{x y}{x-y}\) is all real numbers except when \(x = y\), while the range is all real numbers.

Step by step solution

01

Identifying the Undefined Values

Identify any values of x and y that would make the denominator equal to zero, since division by zero is undefined. This occurs when \(x - y = 0\) or, equivalently, when \(x = y\). Hence, the function is undefined when \(x = y\).
02

Describing the Domain

Considering the limitation derived from the previous step, the domain of the function \(z=\frac{x y}{x-y}\) is all real numbers except for the condition \(x = y\). This can be written as \((x, y) \in \mathbb{R}^2, x \neq y\) where \(\mathbb{R}^2\) denotes a 2D plane of real numbers.
03

Describing the Range

For the range, notice that there are no restrictions on the z values in \(\frac{x y}{x-y}\), so the range of the function is all real numbers \(\mathbb{R}\).

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Most popular questions from this chapter

Find \(d w / d t\) (a) using the appropriate Chain Rule and (b) by converting \(w\) to a function of \(t\) before differentiating. \(w=x y \cos z, \quad x=t, \quad y=t^{2}, \quad z=\arccos t\)

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