Problem 4 - Algebra and Trigonometry

Question

\(1-6=\) 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

The expression evaluates to \(-\frac{1}{3}\) at \(t=1\) and its domain is all real numbers except \(t \neq -2\).

Step 1: Understanding the Expression

The expression given is \(\frac{2t^2 - 5}{3t + 6}\). We need to evaluate this expression when \(t = 1\) and find its domain.

Step 2: Evaluating the Expression at t = 1

Substitute \(t = 1\) into the expression: \[\frac{2(1)^2 - 5}{3(1) + 6} = \frac{2(1) - 5}{3 + 6} = \frac{2 - 5}{9} = \frac{-3}{9} = -\frac{1}{3}.\]

Step 3: Finding the Domain of the Expression

The domain of the expression \(\frac{2t^2 - 5}{3t + 6}\) consists of all real numbers except where the denominator is zero. Set the denominator \(3t + 6\) equal to zero and solve for \(t\):\[3t + 6 = 0 \ 3t = -6 \ t = -2.\]Thus, the domain is all real numbers except \(t = -2\), which can be expressed as \(t eq -2\).

Key Concepts

Expression Evaluation

Evaluating expressions involves substituting a given value for a variable and simplifying the expression to find the result. It's a common process in algebra that allows us to find the value of an expression when one or more variables are known.

In this case, the expression is \( \frac{2t^2 - 5}{3t + 6} \), and we need to evaluate it for \( t = 1 \). To do this, substitute the number 1 for \( t \) in the expression. The expression becomes \( \frac{2(1)^2 - 5}{3(1) + 6} \).

• First, calculate the numerator: \( 2(1)^2 - 5 = 2(1) - 5 = 2 - 5 = -3 \).
• Next, calculate the denominator: \( 3(1) + 6 = 3 + 6 = 9 \).
• Finally, simplify the fraction: \( \frac{-3}{9} = -\frac{1}{3} \).

Thus, the value of the expression when \( t = 1 \) is \( -\frac{1}{3} \).

Domain of a Function

The domain of a function refers to the set of all possible input values (or \( t \) values in this case) for which the function is defined. For algebraic fractions, finding the domain involves identifying values that could make the denominator zero, as division by zero is undefined.

For the expression \( \frac{2t^2 - 5}{3t + 6} \), we focus on the denominator \( 3t + 6 \). Set this equal to zero and solve for \( t \):

• Start with \( 3t + 6 = 0 \).
• Subtract 6 from both sides, resulting in \( 3t = -6 \).
• Divide both sides by 3 to get \( t = -2 \).

Thus, the value \( t = -2 \) makes the denominator zero, and the expression is undefined at this point. Therefore, the domain of \( \frac{2t^2 - 5}{3t + 6} \) is all real numbers except \( t = -2 \), often written as \( t eq -2 \).

Algebraic Fractions

Algebraic fractions are similar to numeric fractions but include variables in either the numerator, denominator, or both parts of the fraction. Working with these expressions requires a solid understanding of both fractions and algebra.

When dealing with algebraic fractions, here are some key points to consider:

• Simplifying: Always try to simplify the fraction by factoring both the numerator and denominator, if possible. This can sometimes make the evaluation process easier.
• Calculating Values: Substitute known values into the expression and simplify carefully to avoid mistakes.
• Finding the Domain: Remember to check where the expression is undefined by looking at the denominator.

Understanding these aspects will help you manage algebraic fractions more efficiently in any mathematical context. Whether you are evaluating them or determining their domain, a meticulous approach is essential to achieve accurate results.