Problem 35 - 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 \(x .\) $$ x y=9 $$

Step-by-Step Solution

Answer

Yes, \( y \) is a function of \( x \) for \( x \neq 0 \).

Step 1: Express y in Terms of x

To determine if the equation defines \( y \) as a function of \( x \), we need to express \( y \) explicitly in terms of \( x \). Let's rearrange the given equation \( xy = 9 \) to solve for \( y \):\[ y = \frac{9}{x} \]

Step 2: Determine if y is a Function of x

A relation is a function if each \( x \) corresponds to exactly one \( y \). From \( y = \frac{9}{x} \), for any non-zero value of \( x \), there is exactly one corresponding value of \( y \). Therefore, \( y \) is a function of \( x \) for all \( x eq 0 \).

Step 3: Check for Any Values of x with Multiple y

Since \( y = \frac{9}{x} \) gives a unique \( y \) for each \( x eq 0 \), there is no single \( x \) value which gives multiple \( y \) values in this context. However, the function is undefined at \( x = 0 \), but this does not affect the function status at non-zero values.

Key Concepts

Relation

In mathematics, a relation is a set of ordered pairs. A relation describes how elements from one set, called the domain, correspond to elements in another set, called the range. The idea is akin to a pairing or connection between two elements.

A relation is considered a function if every element in the domain is associated with exactly one element in the range. When students analyze relations, they check whether a specific input produces only one output. If multiple outputs are produced for the same input, then it's not a function.

For example, in the scenario involving the equation \( xy = 9 \), you're looking to determine if each input value of \( x \) results in exactly one output value of \( y \), which means it's a functional relation.

Equation

An equation is like a mathematical statement that shows the equality between two expressions. Often, equations involve the use of variables, which represent unknown values.

In the exercise involving \( xy = 9 \), this equation is telling us that when you multiply \( x \) and \( y \), the product should be 9.

• One goal when working with equations is to solve for one variable in terms of the others, if possible.
• This means rearranging the equation to express one variable as an explicit function of another.
• For instance, from \( xy = 9 \), solving for \( y \) gives us \( y = \frac{9}{x} \).

Equations form the backbone of many mathematical problems and are crucial in determining whether relations are functions.

Variables

Variables are symbols used to represent unknown numbers in mathematical equations and expressions. They can take on various values, depending on the expressions and equations they're part of.

• In our equation \( xy = 9 \), \( x \) and \( y \) are variables.
• These variables can change and take different values while maintaining the truth of the equation.
• Variables allow us to describe and work with general situations instead of dealing solely with fixed numbers.

Understanding the role of variables is fundamental when determining if an equation represents a function, as you analyze if each possible value of one variable pairs with just one value of another.

Ordered Pairs

Ordered pairs are essentially pairs of numbers or variables placed in a specific sequence, usually written in the form \((x, y)\). These are essential in discussing relations in mathematics.

They represent the specific connections between two sets of values, showing one from the domain and one from the range.

• In functions, every \( x \) value should map precisely to one \( y \) value, which you can represent as ordered pairs like \((x, y)\).
• If there are cases where a single \( x \) associates with multiple \( y \) values, it would lead to multiple ordered pairs like \((x, y_1)\) and \((x, y_2)\), which would contradict the definition of a function.

With functions like \( y = \frac{9}{x} \), where each non-zero \( x \) delivers a unique \( y \), the corresponding ordered pairs illustrate a functional relation by showing a one-to-one mapping.