Problem 2 - Elementary Algebra

Question

Find the domain of each of the following equations. Assume that the independent variable is the variable that appears in the expression on the right side of the \("="\) sign. $$ y=\frac{5}{x} $$

Step-by-Step Solution

Answer

Answer: The domain of the function is all real numbers except for x = 0, which can be written as \(\{x | x \in \mathbb{R}, x \ne 0\}\).

Step 1: Identify Possible Restrictions on the Domain

In the given equation, \(y=\frac{5}{x}\), the function will be undefined if the denominator (x) is equal to zero. This is because division by zero is undefined. So, we want to find the values of x such that \(x \ne 0\).

Step 2: Determine the Domain of the Function

Since the function is only undefined when the denominator is zero, the domain of the function will include all real numbers except for \(x = 0\). Therefore, the domain of the function can be written as: Domain = \(\{x | x \in \mathbb{R}, x \ne 0\}\)

Key Concepts

Independent variable

The independent variable in a function is the variable that you can freely change without being affected by other variables in the function. It is typically represented along the x-axis in graphs.

• In the math equation given, \(y = \frac{5}{x}\), the independent variable is \(x\) because \(x\) is the value you input into the function to determine \(y\).
• An independent variable acts as the 'cause' where the outcome or the 'effect' is the dependent variable. Here, \(y\) depends on how the value of \(x\) changes.

To understand independent variables better, think of them as the factors you control or provide, and each input might alter the outcome or dependent variable. Hence, the independent variable directly affects the function's output.

Undefined expressions

An undefined expression arises when there's a calculation in mathematics that doesn't have a meaningful result. Such expressions can lead to errors in computations.

• A classic example is division by zero, which is undefined. In the equation \(y = \frac{5}{x}\), the expression becomes undefined when \(x = 0\) because dividing any number by zero doesn't yield a valid result.
• Undefined expressions can also occur with negative roots when dealing with real numbers, as they result in complex rather than real numbers.

Identifying undefined expressions is crucial to determining a function's domain. Any value that makes an expression undefined is typically excluded from the domain, ensuring the function operates under legal mathematical operations.

Real numbers

Real numbers are all the numbers that can be found on the number line. They include all the rational and irrational numbers, providing a complete set of values for understanding and solving mathematical problems.

• Rational numbers include fractions and integers, where a number can be expressed as a ratio of two integers.
• Irrational numbers, like \(\pi\) and \(\sqrt{2}\), cannot be written as simple fractions, but still, they find their place on the number line.

When dealing with the domain of a function like \(y = \frac{5}{x}\), real numbers play an essential role. The domain is all real numbers except those values that make the function undefined, such as \(x = 0\). Real numbers provide a broad framework that helps in understanding the scope of possible values a mathematical function can manipulate.