Problem 80 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$g(a)=\frac{4}{2 a^{2}+3 a}$$

Step-by-Step Solution

Answer

The domain of the function \(g(a) = \frac{4}{2a^2 + 3a}\) is given by \(\{a \in \mathbb{R} : a \neq 0 \textrm{ and } a \neq -\frac{3}{2} \}\).

Step 1: Identify the denominator equation

First, we identify the denominator of the given function \(g(a)\): \[ g(a)=\frac{4}{2 a^{2}+3 a} \] Here, the denominator is \(2 a^{2}+3 a\).

Step 2: Find the values of \(a\) for which the denominator is zero

To find the values of \(a\) that would make the denominator zero, we solve the equation \(2 a^{2}+3 a=0\). We can factor out common elements in the equation as follows: \[ a(2a+3)=0 \] Now, we have two possible cases: \[ a = 0 \quad\textrm{or}\quad 2a+3 = 0 \] The first case has an immediate solution, \(a=0\). For the second case, let's solve for \(a\): \[ 2a = -3 \Rightarrow a = -\frac{3}{2} \]

Step 3: Determine the domain of the function

Therefore, the values of \(a\) for which the denominator is zero are \(a = 0\) and \(a = -\frac{3}{2}\). These values will be excluded from the domain, as the function would be undefined at these points. So, the domain of \(g(a)\) will be: \[ \text{Domain of }g(a) = \{a \in \mathbb{R} : a \neq 0 \textrm{ and } a \neq -\frac{3}{2} \} \] This completes the determination of the domain of the given function.

Key Concepts

Rational functions

Rational functions are an interesting and important class of mathematical expressions. Simply put, they are the ratio of two polynomials. For example, a function of the form \(g(a) = \frac{4}{2 a^2 + 3 a}\) is a rational function. The top part, known as the numerator, is any polynomial, and the bottom part, called the denominator, is also a polynomial.
These functions are called "rational" because they can be expressed as a ratio of polynomials.
Rational functions are common in algebra and calculus, and understanding them is crucial for solving complex equations. Their behavior can vary widely based on the degrees and coefficients of the numerator and denominator.

• The numerator determines the zeros or roots of the function.
• The denominator plays a critical role in defining where the function is undefined.

This makes it important to analyze both parts carefully when studying rational functions.

Denominator

The denominator in a rational function is the part that sits below the line in a fraction, in this case \(2 a^2 + 3 a\). It plays a crucial role in defining the function's characteristics. One key point is that the denominator must not be zero, as division by zero is undefined in mathematics.
When analyzing rational functions, identifying the denominator is always an essential step. It helps to:

• Understand where the function might not be valid.
• Recognize how changes in the denominator can affect the entire function.

The denominator affects the vertical structure of the function's graph. Identifying when the denominator becomes zero helps in determining potential "holes" in the graph or points of discontinuity.
For instance, to find where \(g(a)\) becomes undefined, we set \(2 a^2 + 3 a = 0\) and solve this equation. This process reveals the critical points that must be excluded from the function's domain.

Undefined points

Undefined points in the context of rational functions are values for which the function does not exist. These occur when the denominator equals zero, as division by zero is not possible. To find undefined points in a function like \(g(a) = \frac{4}{2 a^2 + 3 a}\), solving the equation \(2 a^2 + 3 a = 0\) provides the necessary values to exclude from the domain.
The process is simple:

• Factor the equation: \(a(2a+3) = 0\)
• The solutions are \(a = 0\) or \(a = -\frac{3}{2}\).

These solutions are the undefined points of the function, meaning the function cannot take these values. Therefore, when determining the domain of \(g(a)\), one should remember to exclude these values, leaving us with all real numbers except \(a = 0\) and \(a = -\frac{3}{2}\). This understanding ensures clarity in calculating and graphing rational functions.