Problem 10 - Precalculus: Mathematics for Calculus

Question

Find the domain of the expression. $$\frac{2 t^{2}-5}{3 t+6}$$

Step-by-Step Solution

Answer

All real numbers except \( t = -2 \).

Step 1: Identify Denominators

The domain of a rational expression is all real numbers except those that make the denominator zero. Start by identifying the denominator of the expression, which is given as \(3t + 6\).

Step 2: Set Denominator Equal to Zero

To find the values that are not in the domain, set the denominator equal to zero: \(3t + 6 = 0\). Our goal is to solve for \(t\).

Step 3: Solve for t

Solve the equation \(3t + 6 = 0\). Subtract 6 from both sides to get \(3t = -6\). Then divide both sides by 3 to find \(t = -2\).

Step 4: Determine the Domain

The domain includes all real numbers except \(t = -2\), since this value makes the denominator zero.

Key Concepts

Understanding Rational Expressions

A rational expression is similar to a fraction, but instead of just numbers, it includes variables. Typically, it is expressed as the quotient of two polynomials. For example, the given expression \( \frac{2t^2 - 5}{3t + 6} \) is a rational expression. It has a numerator, which is \( 2t^2 - 5 \), and a denominator, which is \( 3t + 6 \).

• Rational expressions can resemble fractions with polynomials on the top and bottom.
• It's important to understand these kinds of expressions to solve them correctly using algebraic methods.

The feature that makes rational expressions unique is their behavior depending on the values you plug into the variable, especially if they make the denominator zero.

The Role of the Denominator

In any fraction or rational expression, the denominator is the part underneath the fraction bar. It tells us how many equal parts the whole is divided into. In rational expressions like \( \frac{2t^2 - 5}{3t + 6} \), the denominator is \( 3t + 6 \).

• If the denominator equals zero, the expression becomes undefined.
• Therefore, finding values that make the denominator zero helps in determining the domain of the expression.

For the expression \( 3t + 6 \), set it to zero to identify problematic values. This is a crucial step to understanding what makes the expression undefined.

Defining Real Numbers in the Context

Real numbers are the set of numbers that include both rational numbers (like 2, 3/4, etc.) and irrational numbers (like \( \sqrt{2} \), \( \pi \), etc.). When dealing with rational expressions, we often consider what real numbers will be in the domain—that is, which values we can substitute for the variable without causing any problems.

• The domain excludes numbers that make the denominator zero because division by zero is not allowed in mathematics.
• In our example, any real number is valid except the value that nullifies the denominator.

Through this, you ensure that the output of your rational expression is always a real number, maintaining its mathematical integrity.

Solving Equations to Find Domain

Solving equations is essential in finding the domain of rational expressions. By setting the denominator equal to zero, we determine which values are not allowed. For instance, let's solve the equation \( 3t + 6 = 0 \). This helps us uncover which values make the expression undefined.

• First, subtract 6 from both sides to get \( 3t = -6 \).
• Next, divide both sides by 3, giving \( t = -2 \).

This means all real numbers are in the domain except \( t = -2 \) since plugging this number in turns the denominator into zero, which leads to division by zero and an undefined expression. Finding and excluding these values ensures the expression remains valid throughout its domain.