Problem 22 - A Graphical Approach to College Algebra

Question

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log \left(\frac{x+1}{x-5}\right)$$

Step-by-Step Solution

Answer

Domain: \((-1, 5) \cup (5, \infty)\).

Step 1: Understand Logarithm Domain

The function is a logarithm, \(y = \log\left(\frac{x+1}{x-5}\right)\), and for a logarithm function \(\log(a)\) to be defined, the argument \(a\) must be positive. Thus, \(\frac{x+1}{x-5} > 0\).

Step 2: Solve Inequality for Positive Solutions

Solve \(\frac{x+1}{x-5} > 0\). The quotient is positive when both \(x+1\) and \(x-5\) are either positive or negative.

Step 3: Analyze Signs for Numerator and Denominator

Evaluate the inequality:- \(x+1 > 0\) (i.e., \(x > -1\))- \(x-5 > 0\) (i.e., \(x > 5\))So, \(x > 5\) for both positive.For both negative:- \(x+1 < 0\) (i.e., \(x < -1\))- \(x-5 < 0\) (i.e., \(x < 5\))Thus, \(-1 < x < 5\) for both negative.

Step 4: Combine Intervals for Domain

Combine results from Step 3: \(x > 5\) or \(-1 < x < 5\). The domain of the function is the union of these intervals: \((-1, 5) \cup (5, \infty)\).

Step 5: Graphical Verification (Optional)

Consider plotting \(y = \log\left(\frac{x+1}{x-5}\right)\) to verify. The graph should be defined and recognizable in \((-1, 5) \cup (5, \infty)\) with vertical asymptotes at \(-1\) and \(5\), confirming the analytic findings.

Key Concepts

Domain of a Function

When we talk about the domain of a function, we are asking which values of the variable (usually 'x') will make the function work without any problems. For a logarithmic function, finding the domain is slightly different because it requires the argument of the logarithm to be positive.
The function in the exercise is expressed as \( y = \log \left( \frac{x+1}{x-5} \right) \). The argument here is \( \frac{x+1}{x-5} \), so we need to determine where this fraction is greater than zero. This is crucial because a logarithm of a negative number, or zero, is undefined in the real number system.
To find which values of 'x' make \( \frac{x+1}{x-5} > 0 \), we must analyze how both the numerator \( x+1 \) and the denominator \( x-5 \) behave. The fraction will be positive if both parts are positive or both are negative, which leads to determining separate conditions. In this problem, we find the domain to be the interval \((-1, 5) \cup (5, \infty)\), which means that the function is defined for all 'x' within these intervals, not including -1 and 5.

Inequalities

Inequalities are mathematical statements that describe the relative size or order of two values, often using symbols like > (greater than), < (less than), \( \geq \) (greater than or equal to), or \( \leq \) (less than or equal to).
When dealing with inequalities in functions, you often have to determine where expressions are positive or negative. For the logarithmic function in our exercise, solving \( \frac{x+1}{x-5} > 0 \) involves discerning where this expression becomes greater than zero. This means that the expression inside the inequality must not cross zero or become negative, as logarithms for those values are not defined.
There are two scenarios here:

• Both the numerator \( x+1 \) and the denominator \( x-5 \) are positive, leading to \( x > 5 \).
• Both are negative, which means \( x < -1 \) and \( x < 5 \) but since \( x < -1 \) is already before 5, we shorten it to \( -1 < x < 5 \).

The solution gives us the intervals \((-1, 5)\) and \((5, \infty)\). It's this method of solving inequalities that helps define the boundary and validity conditions for functions.

Graphical Method

The graphical method in mathematics involves visualizing data or functions to interpret their behavior or properties. This approach can be particularly effective for functions like logarithms, where understanding how outputs vary as inputs change can offer deeper insights beyond algebraic solutions.
For the function \( y = \log \left( \frac{x+1}{x-5} \right) \), plotting the graph can help verify the domain found analytically. By visualizing, you see the parts of the graph that correspond to the identified domain \((-1, 5) \cup (5, \infty)\), confirming the function is defined only within these sections.
The graph will reveal vertical asymptotes at \( x = -1 \) and \( x = 5 \). These asymptotes indicate points where the function is undefined or approaches infinity because as 'x' approaches either -1 or 5, the value of the function becomes extremely large in magnitude.

• Visual Confirmation: Seeing the graph can help students understand why certain 'x' values make the logarithmic function undefined.
• Function Behavior: The plot shows the increasing or decreasing trends, highlighting where the function behaves smoothly, which corresponds to its domain.

This reinforces the analytical steps and confirms the solutions derived for the domain by cross-referencing with visual data.