Problem 21 - A Graphical Approach to College Algebra

Question

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log \left(\frac{x+3}{x-4}\right)$$

Step-by-Step Solution

Answer

Domain: \( x < -3 \) or \( x > 4 \).

Step 1: Understand the Logarithmic Function's Requirements

The function given is \( y = \log \left( \frac{x+3}{x-4} \right) \). For the logarithmic function to be defined, the argument (the expression inside the log) must be positive. This means that \( \frac{x+3}{x-4} > 0 \).

Step 2: Determine When the Fraction is Positive

The fraction \( \frac{x+3}{x-4} \) is positive when both the numerator \( x+3 \) and the denominator \( x-4 \) are either both positive or both negative. This gives us two conditions to analyze separately.

Step 3: Analyze Condition 1: Both Positive

For \( \frac{x+3}{x-4} > 0 \) with both parts positive, the conditions are: \( x+3 > 0 \) and \( x-4 > 0 \). Solving these inequalities gives \( x > -3 \) and \( x > 4 \). The combined solution for both is \( x > 4 \).

Step 4: Analyze Condition 2: Both Negative

For \( \frac{x+3}{x-4} > 0 \) with both parts negative, the conditions are: \( x+3 < 0 \) and \( x-4 < 0 \). Solving these gives \( x < -3 \) and \( x < 4 \). The combined solution for both is \( x < -3 \).

Step 5: Combine the Solutions

Combining the solutions from Step 3 and Step 4, the domain of \( y = \log \left( \frac{x+3}{x-4} \right) \) is \( x < -3 \) or \( x > 4 \). This excludes the region where the expression is zero or undefined (i.e., where \( x eq 4 \)).

Key Concepts

Domain Analysis

The domain of a function encompasses all possible input values (typically, the 'x' values) for which the function is defined. For logarithmic functions like \( y = \log \left( \frac{x+3}{x-4} \right) \), analyzing the domain is crucial because the logarithm is only defined for positive values. Thus, we must ensure that the expression inside the logarithm is greater than zero.
First, we consider the expression \( \frac{x+3}{x-4} \), which is the argument of the log. Our key requirement is that \( \frac{x+3}{x-4} > 0 \). This inequality forms the foundation for determining the domain. We also consider any points that make the denominator zero, as these are not part of the domain either because division by zero is undefined. Therefore, we exclude \( x = 4 \) from the domain.

Inequalities

Inequalities are mathematical expressions that describe the relative size or order of two values. In this context, they help us figure out when \( \frac{x+3}{x-4} \) is positive. We analyze two main situations using inequalities:

• Both numerator and denominator are positive.
• Both numerator and denominator are negative.

For the first situation, we set \( x+3 > 0 \) and \( x-4 > 0 \), which resolve to \( x > -3 \) and \( x > 4 \) respectively. The solution where both are true is \( x > 4 \).
For the second scenario, \( x+3 < 0 \) and \( x-4 < 0 \), leading to \( x < -3 \) and \( x < 4 \). Both are true when \( x < -3 \). Hence, these inequalities give us two intervals where the fraction is positive, directing us to the possible solutions for \( x \).

Function Positivity

The positivity of functions is vital when dealing with logarithms because \( \log(x) \) is only defined for \( x > 0 \). This requirement ensures our function outputs real numbers. Analyzing a function like \( \frac{x+3}{x-4} \), we need to verify that the entire expression inside the log stays positive to make the logarithm valid.
This means we must ensure the specified conditions \( x > 4 \) and \( x < -3 \) hold true to maintain positive values of \( \frac{x+3}{x-4} \). Over these intervals, the function remains positive and is graphically represented in sections not touching or crossing zero. Thus, the comprehensive domain of the function is \( (-\infty, -3) \cup (4, \infty) \), as this domain respects both the positivity requirement and exclusion of undefined points where the denominator equals zero.