Problem 10 - Algebra and Trigonometry

Question

An expression is given. (a) Evaluate it at the given value. (b) Find its domain. $$ \frac{2 t^{2}-5}{3 t+6}, \quad t=1 $$

Step-by-Step Solution

Answer

(a) \(-\frac{1}{3}\), (b) Domain: \(t \neq -2\).

Step 1: Substitute and Evaluate

Substitute the value of \( t = 1 \) into the expression \( \frac{2t^2 - 5}{3t + 6} \). This substitution gives us \( \frac{2(1)^2 - 5}{3(1) + 6} \). Simplifying the numerator, \( 2(1)^2 = 2 \), so \( 2 - 5 = -3 \). Simplifying the denominator, \( 3 + 6 = 9 \). Thus, the expression evaluates to \( \frac{-3}{9} = -\frac{1}{3} \).

Step 2: Determine the Domain of the Expression

The domain of a rational expression is all real numbers except those that make the denominator zero. For the expression \( \frac{2t^2 - 5}{3t + 6} \), the denominator is \( 3t + 6 \). Set it to zero: \( 3t + 6 = 0 \). Solving for \( t \), subtract 6 from both sides to get \( 3t = -6 \), then divide by 3 to find \( t = -2 \). Therefore, the domain of the expression is all real numbers except \( t = -2 \).

Key Concepts

Expression Evaluation

When we talk about evaluating an expression, we're focusing on simplifying it by plugging in given values. Let’s break it down further using the expression from the exercise: \( \frac{2t^2 - 5}{3t + 6} \). In this case, the value of \( t \) is given as 1. What this means is that we need to replace each occurrence of \( t \) with 1. Here's how you do it step by step:

• Substitute \( t = 1 \) into the expression, resulting in \( \frac{2(1)^2 - 5}{3(1) + 6} \).
• First, evaluate the numerator: calculate \( 2(1)^2 = 2 \), then \( 2 - 5 = -3 \).
• Next, evaluate the denominator: \( 3 + 6 = 9 \).
• The expression now reduces to \( \frac{-3}{9} \), which simplifies to \( -\frac{1}{3} \).

By following these steps, you can evaluate expressions by substituting any value given. This method is straightforward, but requires careful arithmetic to ensure accuracy.

Domain of a Function

Understanding the domain of a function is all about knowing which values you can plug into it. With rational expressions like \( \frac{2t^2 - 5}{3t + 6} \), the focus is on preventing division by zero, a situation where the expression is undefined. Here’s how you identify the domain:

• Look at the denominator: \( 3t + 6 \).
• Set the denominator equal to zero: solve \( 3t + 6 = 0 \).
• Subtract 6 from both sides to get \( 3t = -6 \).
• Divide both sides by 3: \( t = -2 \).

The function's domain is all real numbers except where the denominator is zero. Thus, for this expression, \( t eq -2 \). In simple terms, \( t \) can be any number but -2. Knowing how to find the domain is essential when working with functions and ensures that your evaluations remain valid.

Substitution Method

The substitution method is a fundamental tool in mathematics used to solve equations or simplify expressions. When you're given a specific value to substitute, like \( t = 1 \) in our example, it helps to approach the problem systematically:

• Identify where to substitute the given value. Here, find each occurrence of \( t \) in \( \frac{2t^2 - 5}{3t + 6} \).
• Replace every \( t \) with the provided number. This involves careful arithmetic to ensure each replacement is correct.
• Simplify the updated expression by working through the mathematical calculations, just as we did by reducing the numerator \( 2 - 5 \) and the denominator \( 3 + 6 \) after substituting.

Substitution isn't just for evaluation—it's a crucial step in solving more complex algebraic expressions and equations. It's reliable, straight-forward, and allows mathematicians and students alike to break down and solve problems effectively.