Problem 33 - Intermediate Algebra

Question

Determine whether each equation defines \(y\) to be a function of \(x .\) If it does not, find two ordered pairs where more than one value of \(y\) corresponds to a single value of \(x .\) $$ y=\frac{1}{x} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Equation Type

The given equation is \( y = \frac{1}{x} \), which is a rational function where \( y \) is expressed explicitly in terms of \( x \).

Step 2: Understand What Defines a Function

A relation is a function if each input \( x \) has exactly one output \( y \). We must check if, for every \( x \), there is only one corresponding \( y \).

Step 3: Check for Multiple Outputs

In this equation, for a given non-zero value of \( x \), the output \( y = \frac{1}{x} \) is uniquely determined to be one specific value. Since the equation simplifies directly to a single \( y \) for every non-zero \( x \), there are not instances of multiple outputs for the same input.

Step 4: Consider Restrictions on the Domain

For \( x = 0 \), the equation is undefined because division by zero is not possible. However, this does not affect the determination, as the function can be restricted to \( x eq 0 \).

Step 5: Conclude Function Status

Since for all allowed inputs (\( x eq 0 \)), \( y \) is uniquely determined. Therefore, \( y = \frac{1}{x} \) does define \( y \) as a function of \( x \).

Key Concepts

Rational Functions

Rational functions play an important role in algebra and calculus. A rational function is essentially a ratio of two polynomials. In simpler terms, it's a fraction where the numerator and the denominator are polynomials. For instance, the function \( y = \frac{1}{x} \) is a basic example of a rational function with a numerator of 1 and a denominator of \( x \). The behavior of these functions can be quite interesting due to the properties of fractions.

• They are defined by polynomials in the numerator and denominator.
• They can have vertical asymptotes, which occur where the denominator is zero.
• The domain of a rational function excludes values that make the denominator zero.

The function \( y = \frac{1}{x} \) is defined for all real numbers except where the denominator is zero, specifically, where \( x = 0 \). This characteristic leads us to explore domain restrictions further.

Domain Restrictions

In mathematics, the domain of a function is the set of all possible input values (typically \( x \)) that the function can accept. Domain restrictions occur when certain values of \( x \) result in undefined or undesirable outputs, such as division by zero or square roots of negative numbers.
For the function \( y = \frac{1}{x} \), the domain excludes \( x = 0 \). This is because division by zero is undefined, leading to a restriction in the domain. Therefore, the domain of \( y = \frac{1}{x} \) includes all real numbers except \( x = 0 \).

• To find the domain of a rational function, find where the denominator is zero and exclude those values.
• Domain restrictions ensure the function operates within a defined set of inputs.

Understanding domain restrictions is crucial because it helps identify where a function is valid, thus preventing mathematical errors during calculations.

Function Definition

A function in mathematics describes a relationship between two sets, where each input from the first set (domain) corresponds to exactly one output in the second set (range). Determining whether a given relation is a function involves checking if every input has a single, unique output.
With the function \( y = \frac{1}{x} \), each non-zero value of \( x \) produces a unique corresponding output for \( y \). This uniqueness satisfies the definition of a function.

• A key feature of a function is that no input \( x \) can have more than one corresponding \( y \).
• Graphs of functions often pass the "vertical line test," meaning no vertical line intersects the graph at more than one point.

This function passes that test for all permissible \( x \) values, affirming its status as a true mathematical function. By understanding what qualifies as a function, we enable clear and accurate communication of mathematical ideas.