Problem 14 - Algebra for College Students

Question

Find the domain of each rational expression. $$\frac{y+5}{y^{2}+9}$$

Step-by-Step Solution

Answer

The domain is all real numbers.

Step 1: Identify the Rational Expression

The given rational expression is \(\frac{y+5}{y^2+9}\).

Step 2: Understand the Denominator

The domain of a rational expression is all real numbers except those that make the denominator zero. Look at the denominator: \(y^2+9\).

Step 3: Set the Denominator Equal to Zero

Set the denominator equal to zero to find any values that are not in the domain: \(y^2 + 9 = 0\).

Step 4: Solve the Equation

Solve the equation for \(y\). Rearrange to get \(y^2 = -9\). Since the square of a real number cannot be negative, this equation has no real solutions.

Step 5: Determine the Domain

Because there are no real solutions that satisfy \(y^2 + 9 = 0\), there are no restrictions on \(y\). Therefore, the domain is all real numbers.

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. For example, \(\frac{y+5}{y^2+9}\). The key thing to remember is that while the numerator can be any polynomial, the denominator cannot be zero as division by zero is undefined.
Understanding this is crucial when working with rational expressions.

• Identify the numerator and the denominator.
• Check the denominator for values that make it zero.

When you master these steps, you can confidently handle rational expressions.

Domain of Functions

The domain of a function is the set of all possible input values (usually represented by \(x\text{ or } y\)) that the function can accept. For rational expressions, finding the domain involves ensuring that the denominator is not zero.
Here’s what you need to do:

• Identify the denominator of the rational expression.
• Set the denominator equal to zero and solve for the variable.
• Exclude these solutions from the domain, as they would make the expression undefined.

In the exercise example \(\frac{y+5}{y^2+9}\), the denominator is \(y^2+9\). Solving for \(y\) in \(y^2+9=0\) shows there are no real values that make the denominator zero. Hence, the domain is all real numbers.

Solving Equations

Solving equations is a fundamental skill in math. It involves finding the values of the variables that make the equation true. When dealing with rational expressions, this often means finding the values that make the numerator or denominator zero.
Steps to solve:

• Isolate the variable on one side of the equation.
• Simplify both sides if needed.
• Check for any solutions that might be excluded from the domain.

In the example equation \(y^2+9=0\), we solve for \(y\) but find that there are no real solutions because the equation requires the square of a number to be negative, which is impossible for real numbers. Understanding this helps prevent errors and confirms the domain of the function.