Question

Find the domain of the expression.\(\frac{1}{x-2}\)

Short answer

The domain of the expression \(\frac{1}{x-2}\) is all real numbers except 2

Step-by-step solution

Set the denominator equal to zero

To find values that make the denominator zero, take the denominator which is \(x-2\), and set it equal to zero, like this: \(x-2 = 0\).

Solve the equation

Solving the equation \(x-2 = 0\), the value of x is: \(x = 2\). This is the value that makes the denominator zero resulting in the expression being undefined.

Determine the Domain

The domain of an expression is all real values that x can be, except any that would make the expression undefined. Since \(x = 2\) makes the expression undefined, that value that must be excluded from the domain. Therefore, the domain of the expression is all real numbers except 2.

Key concepts

Denominator

In rational expressions like \(\frac{1}{x-2}\), the denominator is the part of the fraction located below the fraction line (the divisor). It plays a crucial role in determining the domain of the expression. For a fraction to be valid, the denominator cannot be zero, as division by zero is undefined in mathematics. Therefore, finding when the denominator equals zero is essential for determining where the expression might not work.

To find this, take the denominator \(x-2\) and set it equal to zero. Solving \(x-2=0\) gets you \(x=2\). This means when \(x=2\), the denominator is zero, leading to a potentially problematic situation for the expression where it cannot effectively operate.

Undefined Expression

An undefined expression occurs when a mathematical expression does not result in a real number. For rational expressions, this often happens when the denominator is zero. Since division by zero is undefined, values of the variable that make the denominator zero must be excluded from the domain. In our case with the expression \(\frac{1}{x-2}\), setting \(x=2\) results in a zero denominator, making the expression undefined. Therefore, \(x=2\) is not valid for the domain of the expression.

Being attentive to undefined expressions is important in understanding the behavior and limitations of rational functions. It's crucial to identify such points so the expression functions correctly with valid values.

Real Numbers

The domain of a mathematical expression refers to all the possible values that the variable(s) can take without making the expression invalid or undefined. For the expression \(\frac{1}{x-2}\), we want the domain to include all real numbers except those that make the denominator zero, which in this case is \(x=2\).

Real numbers include all the numbers that you might find on the number line, including both positive and negative numbers, as well as zero. They encompass rationals, like decimals and fractions, and irrationals, like \(\pi\) and \(\sqrt{2}\). But in our scenario, all real numbers are allowed except where the expression is undefined. Thus, the domain excludes \(x=2\), leading to the domain being all real numbers except this one point.

• This can be written in set notation as \( \{x \in \mathbb{R} \mid x eq 2\} \) or interval notation as \((-\infty, 2) \cup (2, \infty)\).

This ensures the expression functions well wherever it is valid, accommodating all other real number inputs.