Problem 123 - Elementary Algebra

Question

Find the domain of the equations for the following problems. $$ y=\frac{4}{x-2} $$

Step-by-Step Solution

Answer

Question: Determine the domain of the function \(y=\frac{4}{x-2}\). Answer: The domain of the given function is \(x\in(-\infty,2)\cup(2,\infty)\).

Step 1: Identify the Denominator

In the given function \(y=\frac{4}{x-2}\), the expression in the denominator is \((x-2)\). We need to determine the values of x that make this denominator equal to zero.

Step 2: Find the Values That Make the Denominator Zero

To find the values of x that make \((x-2)=0\), we set the denominator equal to 0 and solve for x. $$ x-2=0 $$ Add 2 on both sides: $$ x=2 $$

Step 3: Determine the Domain

The value of x that makes the denominator zero is \(x=2\). The domain of the function is all real numbers except this value. Thus, the domain can be written as: $$ x\in(-\infty,2)\cup(2,\infty) $$

Key Concepts

Rational Functions

Rational functions are a type of function expressed as the ratio of two polynomials. In simple terms, you have a fraction where the numerator and the denominator are both polynomials. For example, in the function \( y = \frac{4}{x-2} \), the numerator is 4 (a constant, which is also a polynomial) and the denominator is \(x-2\), which is a simple linear polynomial.
Rational functions are important in mathematics because they can represent many real-world situations, like rates and ratios. One key characteristic of rational functions is that they can have excluded values, which are numbers that make the denominator zero and make the function undefined. This leads us to the next core concept—excluded values.

Excluded Values

Excluded values are specific numbers that a function cannot accept as inputs, often because they cause a mathematical error such as division by zero. In the context of rational functions, these are the values of \(x\) that make the denominator equal to zero.
In the given function \( y = \frac{4}{x-2} \), if we calculate \(x-2 = 0\), we find that \(x = 2\). This is an excluded value for the function because substituting \(x = 2\) would make the denominator zero, resulting in an undefined expression. Consequently, the domain of the function will include all real numbers except this excluded value.

Real Numbers

Real numbers are all the numbers on the number line, including both rational numbers (such as fractions and integers) and irrational numbers (numbers that cannot be expressed as simple fractions, like \(\pi\) or \(\sqrt{2}\)). Real numbers include both positive and negative numbers, whole numbers, and decimals.
When discussing the domain of a function, especially a rational function, real numbers are crucial since they define the set of input values that won’t cause any mathematical errors. For the example function \( y = \frac{4}{x-2} \), while \(x = 2\) is excluded, all other real numbers are part of the domain. Thus, the domain is written as \(x \in (-\infty, 2) \cup (2, \infty)\), meaning that \(x\) can be any real number except 2. Understanding real numbers helps us grasp the concept of domains in functions, ensuring we only use inputs that keep the function well-defined and meaningful.