Problem 24 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Basic Properties

The function given is a logarithmic function. We know that the expression inside a logarithm must be greater than zero. For our function \(y = \log |6x+6|\), the absolute value \(|6x+6|\) must be greater than zero, i.e., \(|6x+6| > 0\). This condition is vital for determining the domain.

Step 2: Solve Inequality for Absolute Value

Solve the inequality \(|6x+6| > 0\). The absolute value \(|A| > 0\) implies that \(A > 0\) or \(A < 0\). Hence, for \(|6x+6| > 0\), we have two cases: 1. \(6x + 6 > 0\), which simplifies to \(x > -1\).2. \(6x + 6 < 0\), which simplifies to \(x < -1\). Combining these, we note that any real number \(x\) other than \(-1\) satisfies the inequality \(|6x+6| > 0\).

Step 3: Determine the Domain

Based on the inequality solutions, the domain of the function consists of all real numbers except \(-1\). Therefore, the domain is \((-\infty, -1) \cup (-1, +\infty)\).

Step 4: Verify with Graphical Interpretation

The function \(y = \log |6x+6|\) can also be checked graphically by plotting it. You'll observe that there's a vertical asymptote at \(x = -1\), where the function is undefined, confirming the domain \((-\infty, -1) \cup (-1, +\infty)\).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial in mathematics as it tells us which values of the input are allowed. For any given function, the domain consists of all the possible input values that won't break its mathematical rules. When dealing with logarithmic functions, such as the one in our exercise, the expression inside the logarithm must always be greater than zero because the logarithm of zero or a negative number is undefined.

In our case, the function is defined as \( y = \log |6x+6| \). This requires that the expression inside the absolute value function, \(|6x + 6|\), must be greater than zero. Intuitively, this means the values of \(x\) should not make \(6x+6\) equal to zero, which is exactly what happens when \(x = -1\).

Therefore,

• Values where \(|6x+6| = 0\) must be excluded from the domain.
• Any other real number will work as input, thereby the domain is \(( -\infty, -1 ) \cup ( -1, +\infty )\).

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, without considering its direction (negative or positive). It transforms both negative and positive numbers into their positive counterparts. In this exercise, we deal with the absolute value \(|6x + 6|\).

This means that:

• If \(6x+6\) is positive, then \(|6x+6|\) equals \(6x+6\).
• If \(6x+6\) is negative, then \(|6x+6|\) equals \(-(6x+6)\).

The absolute value is essential for defining functions particularly when determining the set of permissible inputs for logarithms. By recognizing how such values behave, we can solve inequalities like \(|6x+6| > 0\), leading us to the valid domain where the logarithm is defined. The solution implies

• \(6x + 6 eq 0\), or equivalently, \(x eq -1\).
This showcases why \(x = -1\) is excluded from the domain.

Solving Inequalities

An inequality establishes a condition where one expression is considered greater or less than another. Solving inequalities allows us to pinpoint which values satisfy these conditions. For the absolute value inequality \(|6x + 6| > 0\), we can simplify to understand where this condition holds true.

Given the inequality \(|6x + 6| > 0\), you break it into two parts:

• \(6x + 6 > 0\), simplifying to \(x > -1\).
• \(6x + 6 < 0\), simplifying to \(x < -1\).

In this context, our solution implies that any \(x\) value except \(-1\) satisfies the condition. Thus, the inequality indicates:

• \(x\) is any real number except \(-1\).

Solving such inequalities is quintessential for determining the domain of logarithmic functions as it reveals input values that produce valid outputs. By doing so, it underscores why the value \(-1\) is omitted from the domain range.