Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, find the domain of each function using interval notation. \(f(x)=\frac{3 x+1}{4 x+2}\)

Step-by-Step Solution

Answer

The domain is \((-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)\).

Step 1: Identify the Potential Domain Restrictions

The domain of a function is all possible input values (usually \(x\) values) that will give a valid output. Since this function is a rational function, the denominator cannot be zero, as division by zero is undefined. Therefore, identify values of \(x\) that make the denominator zero.

Step 2: Set the Denominator Equal to Zero

Write the equation to find where the denominator is zero: \[4x + 2 = 0\] We need to solve this equation to find the value of \(x\) that causes the denominator to be zero.

Step 3: Solve for x

Subtract 2 from both sides of the equation: \[4x = -2\] Then, divide both sides by 4 to isolate \(x\): \[x = -\frac{1}{2}\] This means \(x = -\frac{1}{2}\) is the value that makes the denominator zero, and thus must be excluded from the domain.

Step 4: Write the Domain in Interval Notation

The function is undefined at \(x = -\frac{1}{2}\), so the domain includes all real numbers except \(x = -\frac{1}{2}\). In interval notation, this is written as: \((-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)\).

Key Concepts

Rational Functions

A rational function is a type of function that is expressed as the ratio of two polynomials. In other words, it is a fraction where both the numerator and the denominator are polynomials. An example of a rational function is \(f(x) = \frac{3x+1}{4x+2}\). The value of the function depends on the expression in the numerator and denominator. However, one critical aspect of rational functions is that the denominator can never be zero. This is because division by zero is undefined in mathematics. Hence, determining for which values of \(x\) the denominator equals zero is crucial when dealing with rational functions.

Interval Notation

Interval notation is a way of representing a set of numbers along a number line. It succinctly expresses the domain of a function or any subset of real numbers. Instead of listing all numbers, interval notation uses brackets or parentheses to describe the start and stop points of the interval.

- A parenthesis \(()\) indicates that an endpoint is not included in the interval, known as an open interval.- A square bracket \([]\) signifies that an endpoint is included, known as a closed interval.

In our example, we use interval notation to exclude the point where the rational function is undefined due to the denominator being zero. Therefore, the domain is represented as \((-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)\). This means all real numbers are included except \(x = -\frac{1}{2}\)."

Denominator Restrictions

In rational functions, denominator restrictions are vital as they determine the values of \(x\) that need to be excluded from the domain of the function. The primary restriction comes from the fact that the denominator, being a polynomial, cannot be zero. To find these restrictions, follow these steps:

• Set the denominator equal to zero. In our example, this means solving the equation \(4x + 2 = 0\).
• Solve for \(x\) to get the specific values that make the denominator zero, leading to undefined points in the function.

For \(f(x) = \frac{3x+1}{4x+2}\), the denominator becomes zero at \(x = -\frac{1}{2}\). These values are omitted from the domain, ensuring that the rational function remains valid for all allowable inputs.