Problem 20 - A Graphical Approach to College Algebra

Question

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log \left(x^{3}-81 x\right)$$

Step-by-Step Solution

Answer

The domain is \((-9, 0) \cup (9, \infty)\).

Step 1: Understand the Logarithmic Function

The function given is \( y = \log{(x^3 - 81x)} \). The logarithm is only defined for positive arguments, which means the expression \( x^3 - 81x \) must be greater than zero: \( x^3 - 81x > 0 \).

Step 2: Factor the Inequality

To solve \( x^3 - 81x > 0 \), we first factor the expression: \( x^3 - 81x = x(x^2 - 81) = x(x - 9)(x + 9) \). This gives us three factors: \( x \), \( x - 9 \), and \( x + 9 \).

Step 3: Identify Critical Values

The critical points occur when any factor equals zero: \( x = 0 \), \( x = 9 \), or \( x = -9 \). These are potential boundaries between intervals where the product changes sign.

Step 4: Test Intervals for Positivity

The critical values divide the real line into intervals: \((-\infty, -9)\), \((-9, 0)\), \((0, 9)\), and \((9, \infty)\). Test each interval:- For \((-\infty, -9)\), choose \(x = -10\): \( (-10)(-19)(-1) = -190 < 0 \).- For \((-9, 0)\), choose \(x = -1\): \( (-1)(-10)(8) = 80 > 0 \).- For \((0, 9)\), choose \(x = 1\): \( 1(-8)(10) = -80 < 0 \).- For \((9, \infty)\), choose \(x = 10\): \( 10(1)(19) = 190 > 0 \).The expression is positive for intervals \((-9, 0)\) and \((9, \infty)\).

Step 5: Write the Domain

The domain of the function is where the logarithmic argument is positive: \((-9, 0) \cup (9, \infty)\). Thus, the domain in interval notation is \((-9, 0) \cup (9, \infty)\).

Key Concepts

Domain of a Function

The domain of a function represents all the possible input values (or 'x' values) that the function can accept without resulting in undefined or non-real outputs. For logarithmic functions, like in our problem, the domain is specifically controlled by the requirement that the argument of the logarithm must be strictly greater than zero. This is because the logarithm of zero or a negative number is undefined in real number mathematics.

Consider the function given: \[ y = \log(x^3 - 81x) \]Here, the expression \( x^3 - 81x \) inside the log must be positive for the logarithm to be defined. Therefore, we need to solve the inequality:\[ x^3 - 81x > 0 \]To find the domain, we end up exploring the values that satisfy this inequality. This means understanding the polynomial itself and finding the intervals over which it is positive.

Start by factoring the expression to find when it equals zero, giving critical points which serve as boundaries for testing intervals where the function is defined. Understanding the domain is crucial for knowing when a function is correctly plotted on the graph or used in calculations.

Critical Points

Critical points in mathematics are specific values in the domain of a function where the behavior or 'direction' of the function changes. These are typically places where the function equals zero or where its derivative equals zero or is undefined. In the context of setting the argument inside a logarithmic function as positive, finding critical points aids in dividing the domain into manageable intervals.

For the given function, the critical points are the solutions to the equation:\[ x(x - 9)(x + 9) = 0 \]Solving this, we find:

• \(x = 0\)
• \(x = 9\)
• \(x = -9\)

These points segment the real number line into different intervals. Each of these intervals needs to be tested to determine the sign of the product, which shows when the function is positive, hence determing part of the domain.

Thus, these critical point calculations are foundational in simplifying and solving inequalities regarding polynomial expressions, especially for nonlinear equations often seen within functions like logarithms.

Interval Testing

Interval testing is a pivotal step in analyzing the parts of a function's domain that keep it defined, or behavior like positivity or negativity of polynomial expressions. For our logarithmic function, interval testing helps us understand how the sign of \(x^3 - 81x\) changes across different ranges, allowing us to precisely determine the domain.

Once the critical points are identified, the entire number line is split into intervals for testing:

• \((-\infty, -9)\)
• \((-9, 0)\)
• \((0, 9)\)
• \((9, \infty)\)

You'll pick a test point from each interval and substitute it into the factored expression \(x(x - 9)(x + 9)\). For example, selecting \(x = -10\) in \((-\infty, -9)\) results in a negative product, meaning our logarithmic function isn’t defined there, while \(x = 10\) in \((9, \infty)\) gives a positive value, meaning that's part of the domain.

Interval testing, even though a straightforward concept, is a powerful tool in analyzing the behavior of polynomial expressions across their domains, particularly for resolving practical problems in calculus and algebra involving inequalities.