Problem 73 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$R(t)=-\frac{t-4}{7 t+3}$$

Step-by-Step Solution

Answer

The domain of the function \(R(t) = -\frac{t-4}{7t+3}\) is all real numbers except for \(t = \frac{-3}{7}\). In set notation, the domain is \( \{ t \in \mathbb{R} : t \neq \frac{-3}{7} \}\) and in interval notation, the domain is \((-\infty, -\frac{3}{7}) \cup (-\frac{3}{7}, +\infty)\).

Step 1: Identify the function's denominator

We will consider the function given: \[R(t) = -\frac{t-4}{7t+3}\] The denominator of this function is \(7t+3\).

Step 2: Determine the values that make the denominator zero

Now we will find the values of \(t\) that make the denominator equal to zero: \[7t + 3 = 0\]

Step 3: Solve for t

To solve for \(t\) in the equation found in step 2, we will isolate \(t\): \[7t = -3\] \[t = \frac{-3}{7}\] The value \(t = \frac{-3}{7}\) makes the denominator of the function equal to zero.

Step 4: Determine the domain

Knowing that the function is undefined when the denominator is zero, we can now determine the domain. The domain of the function R(t) consists of all possible values of \(t\), except for the value that makes the denominator zero. Therefore, the domain of the function R(t) is all real numbers except for \(t = \frac{-3}{7}\). Set notation: \[Domain = \{ t \in \mathbb{R} : t \neq \frac{-3}{7} \}\] Interval notation: \[Domain = (-\infty, -\frac{3}{7}) \cup (-\frac{3}{7}, +\infty)\]

Key Concepts

Function Denominator

In any rational function, the denominator plays a crucial role. A rational function is a fraction where both the numerator and the denominator are polynomials. For the function \( R(t) = -\frac{t-4}{7t+3} \), the denominator is \( 7t+3 \). This part of the function is extremely important because the values that make the denominator zero must be excluded from the domain of the function. To find these values, you simply have to solve for when the denominator equals zero. Thus, identifying the denominator is the first essential step for discovering where a function might be undefined.

Domain Exclusion

A key part of determining the domain of a rational function is finding where the function is undefined due to the denominator equaling zero. When we encounter a term like \( 7t+3 \), we set it equal to zero to find the domain exclusions:

• Set the equation: \( 7t + 3 = 0 \)
• Solve for \( t \) to get \( t = -\frac{3}{7} \)

When \( t = -\frac{3}{7} \), the denominator becomes zero, making the function undefined. Therefore, we exclude \( t = -\frac{3}{7} \) from the domain. Domain exclusions are crucial because they ensure that the function remains valid for all other values.

Rational Function

Understanding rational functions is pivotal in mathematics. A rational function comprises a numerator and a denominator, with both being polynomials. They can model various real-world scenarios but come with the caveat that the denominator should never be zero. The rational function \( R(t) = -\frac{t-4}{7t+3} \) serves as a fundamental example. It highlights the need to consider restrictions on the variable \( t \) to avoid undefined behaviors. This ensures that calculations involving the function yield a real number and prevents mathematical errors.

Set and Interval Notation

Expressing the domain of a function can be done in two primary ways: set notation and interval notation. These notations help clearly communicate which values the variable can take. For \( R(t) = -\frac{t-4}{7t+3} \), the domain avoids the point where the denominator is zero.- **Set notation** displays the domain as: \[ \{ t \in \mathbb{R} : t eq -\frac{3}{7} \} \] - **Interval notation** provides an alternative visual, with: \[ (-\infty, -\frac{3}{7}) \cup (-\frac{3}{7}, +\infty) \] By using these notations, mathematicians can easily and precisely define all valid inputs for a function, ensuring that anyone reading the expression understands its limits.