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Chapter 2: Problem 71

Determine whether each relation defines \(y\) as a function of \(x\). \(x y=1\)

Short Answer

Expert verified
Yes, \(y = \frac{1}{x}\) defines \(y\) as a function of \(x\) for \(x eq 0\).

Step by step solution

01

Understand the given relation

The relation given is \(xy = 1\). We need to determine whether this defines \(y\) as a function of \(x\).
02

Solve for y

Rearrange the equation to solve for \(y\): \[ y = \frac{1}{x} \]
03

Check for function definition

Recall that a relation is a function if every input \(x\) has exactly one output \(y\). Given \[ y = \frac{1}{x} \], for each value of \(x\) (except \(x=0\)), there is a unique corresponding value of \(y\).
04

Consider the domain

Note that \(x\) cannot be zero since division by zero is undefined. However, in all other cases, \(y\) is uniquely determined by \(x\).
05

Conclusion

Since for every \(x eq 0\) there is a unique \(y\), the relation \(xy=1\) defines \(y\) as a function of \(x\) for \(x eq 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Relation Definition
A relation is a kind of rule that links elements of one set, called the domain, with elements of another set, called the range. Each pairing between an input and an output forms a relation. In our exercise, the relation given is expressed as the equation \( xy = 1 \). Understanding this equation in the context of relations is key in determining whether it is also a function.
Solving for y
To determine whether our relation defines \(y\) as a function of \(x\), we start by solving for \(y\). This step involves rearranging the given equation \( xy = 1 \) to isolate \( y \). By doing so, we get: \ ( y = \frac{1}{x} ). Now we have expressed \( y \) directly in terms of \( x \), which helps us analyze the relationship between the two variables more easily.
Unique Output
In mathematics, a relation is considered a function if each input (or value of \(x\)) corresponds to exactly one output (value of \(y\)). For our equation \( y = \frac{1}{x} \), for every allowable value of \( x \), there is a unique value of \( y \). For example, if \( x = 2 \), then \( y = \frac{1}{2} \). Similarly, if \( x = -3 \), then \( y = -\frac{1}{3} \). This uniqueness in the output for each input is confirmed, except where division by zero is involved.
Function Domain
The domain of a function is the complete set of possible values of the independent variable (typically \( x \)). In our equation \( y = \frac{1}{x} \), we must exclude zero from the domain because division by zero is undefined. Therefore, the domain of our function includes all real numbers except zero. This is often written in interval notation as \( (-\infty, 0) \cup (0, \infty) \). Despite this restriction, we can confirm that the relation \( xy = 1 \) does define \( y \) as a function of \( x \) for \( x e 0 \).

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