Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$f(x)=\log |6 x+6|$$

Step-by-Step Solution

Answer

The domain is \((-\infty, -1) \cup (-1, \infty)\).

Step 1: Understanding the Logarithmic Function

The function given is \(f(x) = \log|6x + 6|\). A logarithmic function is defined only for positive arguments. This means that inside the logarithm, the expression \(|6x + 6|\) must be greater than 0.

Step 2: Solving the Inequality

To ensure the argument of the logarithm is positive, set the inequality: \(|6x + 6| > 0\). This inequality implies that \(6x + 6 eq 0\). To solve this, isolate \(x\): \[6x + 6 = 0 \] \ Solve for \(x\) by subtracting 6 from both sides: \[6x = -6 \] \ Divide both sides by 6: \[x = -1 \] Thus, \(6x + 6\) is non-zero when \(x eq -1\). The domain is therefore all \(x\) except \(x = -1\).

Step 3: Writing the Domain

Since \(x\) can be any real number except \(-1\), the domain of \(f(x) = \log|6x + 6|\) is: \((-\infty, -1) \cup (-1, \infty)\).

Key Concepts

Domain of a Function

When we talk about the domain of a function, we are trying to find all the possible input values for which the function is defined. For logarithmic functions, the argument inside the logarithm (what you're taking the log of) must be greater than zero. This is because logarithms of non-positive numbers are not defined in the real number system.

In our specific function, \(f(x) = \log|6x + 6|\), the absolute value \(|6x + 6|\) needs to be greater than zero. This means we solve the inequality \(|6x + 6| > 0\).

• This simplifies to \(6x + 6 eq 0\), which helps us identify values that \(x\) cannot take.
• When we solve \(6x + 6 = 0\), we find \(x = -1\).

Therefore, the domain of the function is all real numbers except \(x = -1\), written in interval notation as \((-\infty, -1) \cup (-1, \infty)\). It's important to exclude this value to maintain the function's validity.

Inequalities

Inequalities are expressions that define the range of values that satisfy a given condition. In the case of our logarithmic function, the condition is that the argument of the log function must be greater than zero.

To explore this further, we solved the inequality \(|6x + 6| > 0\). Absolute values can make things a bit tricky, as they reflect both positive and negative scenarios. However, in this instance, maintaining the expression simply different from zero suffices:

• Finding \(6x + 6 eq 0\) leads us to \(x eq -1\).
• This means the inequality holds for all \(x\) except where \(x\) is precisely \(-1\).

So, inequalities help show us the range of acceptable values \(x\) can take without breaching the condition. Recognizing these values maintains the function's definition across its domain.

Graphical Methods

Graphical methods offer a visual representation of mathematical functions, helping us understand where a function is defined and continuous. When dealing with logarithmic functions and their domains, a graph can illustrate where the function is valid—highlighting restrictions on \(x\).

For the function \(f(x) = \log|6x + 6|\), a graph helps visually assert the domain we calculated analytically. Here's how:

• Plotting the expression \(|6x + 6| > 0\) on a graph, you’ll see the function dips at \(x = -1\), confirming it’s undefined there.
• The graph visually shows where the function rises and where restrictions apply, solidifying our understanding of the domain \((-\infty, -1) \cup (-1, \infty)\).

Graphical methods serve as an intuitive check to confirm the analytical steps taken, ensuring our solutions are robust and comprehensive.