Problem 11 - Algebra for College Students

Question

Find the domain of each rational expression. $$\frac{5 y-1}{y^{2}-4}$$

Step-by-Step Solution

Answer

All real numbers except \(y = 2\) and \(y = -2\).

Step 1: Identify the Rational Expression

The given rational expression is \(\frac{5y - 1}{y^2 - 4}\).

Step 2: Determine the Denominator

The denominator of the rational expression is \(y^2 - 4\).

Step 3: Find Values That Make the Denominator Zero

Set the denominator equal to zero: \(y^2 - 4 = 0\).

Step 4: Solve for \(y\)

Solve the equation \(y^2 - 4 = 0\) for \(y\): \[ y^2 - 4 = (y - 2)(y + 2) = 0. \] So, \(y = 2\) or \(y = -2\).

Step 5: State the Domain

The values that make the denominator zero are \(y = 2\) and \(y = -2\). Therefore, the domain excludes these values. The domain of the rational expression is all real numbers except \(y = 2\) and \(y = -2\). In interval notation, the domain is: \((-∞, -2) \cup (-2, 2) \cup (2, ∞)\).

Key Concepts

Rational Expression

A rational expression is a fraction where both the numerator and the denominator are polynomials. For example, in the expression \(\frac{5y - 1}{y^2 - 4}\), the numerator is \(5y - 1\) and the denominator is \(y^2 - 4\). Understanding rational expressions is crucial because one needs to find values that make the denominator zero to understand the domain. Since division by zero is undefined, identifying these zero values helps in determining the domain of the expression.

Rational expressions are commonly used in algebra, and handling them correctly is important for solving equations involving fractions polynomials.

Denominator

The denominator is the bottom part of a fraction in a rational expression. It plays a critical role because it cannot be equal to zero, as division by zero is impossible. In our example, the denominator is \(y^2 - 4\).

To find dangerous values (or values that make the denominator zero), you need to solve the equation where the denominator is set to zero. For \(y^2 - 4\), you solve \(y^2 - 4 = 0\). This breaks down to \((y-2)(y+2) = 0\), meaning \(y = 2\) or \(y = -2\) makes the denominator zero.

This exact determination prevents these values from being in the domain.

Interval Notation

Interval notation is a shorthand way of writing the domain of a rational expression. It shows which values are included or excluded from a set. For our example, after determining the problematic values of the denominator (\(y = 2, -2\)), we need to exclude these from the domain.

The domain is all real numbers except \(2\) and \(-2\). So, in interval notation, this is expressed as: \((-∞, -2) \cup (-2, 2) \cup (2, ∞)\).

This means all values from negative infinity to -2, from -2 to 2, and from 2 to positive infinity are included, but \(2\) and \(-2\) are not included.

Excluded Values

Excluded values in the domain of a rational expression are the values that make the denominator zero. These values are identified by solving the denominator set equal to zero.

In our example \(\frac{5y - 1}{y^2 - 4}\), setting \(y^2 - 4\) equal to zero gives the solutions \(y = 2\) and \(y = -2\). Thus, \(2\) and \(-2\) are excluded from the domain, as they make the denominator zero and the expression undefined.

Recognizing these values ensures you correctly establish the valid domain for rational expressions, avoiding mathematical errors.