Problem 67 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$f(x)=\frac{6}{x+8}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \frac{6}{x + 8}\) is \(x \in \mathbb{R}, x \neq -8\).

Step 1: Identify the function and its denominator

We are given the function \(f(x) = \frac{6}{x + 8}\). It is a rational function with a denominator of \(x + 8\).

Step 2: Find the values of x that make the denominator zero

We have to find x-values where the denominator equals zero, so we will set the denominator equal to zero and solve for x. \(x + 8 = 0\) Subtract 8 from both sides: \(x = -8\) Thus, the function is not defined at \(x = -8\).

Step 3: Write down the domain of the function

The domain is all real numbers except for the value that makes the denominator equals zero (x = -8). Therefore, the domain of the function is: \(x \in \mathbb{R}, x \neq -8\).

Key Concepts

Rational Functions

Rational functions are a type of function represented by a fraction. They have a polynomial in the numerator and the denominator. This is different from simpler fractions, where both the numerator and the denominator are constants or whole numbers. For example, in the function \(f(x) = \frac{6}{x+8}\), the numerator is the constant 6, and the denominator is \(x+8\), a polynomial of degree 1.
The main thing to remember about rational functions is that they allow us to model many scenarios in mathematics and real life that exhibit a ratio (which involves division). However, the presence of a variable in the denominator introduces additional considerations when determining the function's domain.

Denominator Restrictions

One key aspect of dealing with rational functions is handling the restrictions posed by the denominator. The denominator of a rational function cannot be zero because division by zero is undefined. This means we must find any values of \(x\) that would make the denominator zero and exclude them from the domain.
Let's take \(f(x) = \frac{6}{x+8}\) as an example. The denominator here is \(x+8\). To find the restriction, we set \(x+8 = 0\) and solve for \(x\). By doing this, we find that \(x = -8\) would make the denominator zero. Therefore, \(x = -8\) is a value that the function cannot accept, leading to the need for its exclusion from the domain.

Real Numbers Exclusion

When determining the domain of a rational function, understanding how to exclude certain real numbers is vital. The domain concept in mathematics refers to all possible input values (\(x\)-values) that a function can accept. Rational functions often start with the assumption that they can accept all real numbers, but because of denominator restrictions, certain values are excluded.
In our example, we determined that \(x = -8\) causes the denominator to become zero, which is not allowed. So, the domain of \(f(x) = \frac{6}{x+8}\) is all real numbers except \(x = -8\). This can be expressed in interval notation as \((-\infty, -8) \cup (-8, \infty)\), meaning \(x\) can take any real number value except for \(-8\). Understanding how to exclude specific numbers from the domain is a critical skill when working with rational functions.