Problem 19 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The domain is \((-1, 0) \cup (1, \infty)\)."

Step 1: Understand the Function

The function given is \( y = \log(x^3 - x) \). For the logarithm to be defined, the expression inside the logarithm must be positive, i.e., \( x^3 - x > 0 \). This ensures that the logarithm is defined and real.

Step 2: Solve Inequality for Domain

Solve the inequality \( x^3 - x > 0 \) by factoring the expression. The equation can be rewritten as \( x(x^2 - 1) > 0 \). Further factor this as \( x(x-1)(x+1) > 0 \). Find the critical points by setting each factor to zero: \( x = 0 \), \( x = 1 \), and \( x = -1 \).

Step 3: Test Intervals Around Critical Points

The critical points divide the number line into intervals: \((-\infty, -1)\), \((-1, 0)\), \((0, 1)\), and \((1, \infty)\). Test a value from each interval to determine if the product \( x(x-1)(x+1) > 0 \).

Step 4: Determine Valid Intervals

- For \((-\infty, -1)\), test \( x = -2 \): \((-2)((-2)-1)((-2)+1) < 0\), so it's not part of the domain. - For \((-1, 0)\), test \( x = -0.5 \): \((-0.5)((-0.5)-1)((-0.5)+1) > 0\), so it is part of the domain.- For \((0, 1)\), test \( x = 0.5 \): \((0.5)((0.5)-1)((0.5)+1) < 0\), so it's not part of the domain.- For \((1, \infty)\), test \( x = 2 \): \((2)((2)-1)((2)+1) > 0\), so it is part of the domain.

Step 5: Final Domain Statement

The intervals where \( x(x-1)(x+1) > 0 \) are \((-1, 0)\) and \((1, \infty)\). Thus, the domain of the function is \((-1, 0) \cup (1, \infty)\).

Key Concepts

Critical Points

The concept of critical points is essential in determining where the behavior of a function changes. In our exercise, to find the domain of the logarithmic function \( y = \log(x^3 - x) \), we first set the expression within the logarithm to greater than zero, \( x^3 - x > 0 \). To proceed further, we factor this to \( x(x^2 - 1) > 0 \), and then further into \( x(x-1)(x+1) > 0 \).

The zeros of these factors—where each factor equals zero—are known as critical points. For our function, the critical points are where \( x = 0 \), \( x = 1 \), and \( x = -1 \). These points are crucial because they divide the number line into intervals. Each interval might have a different sign (positive or negative) when substituted back into the original inequality.

• These critical points help us test and discover which intervals satisfy our condition \( x(x-1)(x+1) > 0 \).
• After identifying these intervals, we determine whether the product is positive, as required for the domain of the logarithmic function.

Inequalities in Algebra

Inequalities in algebra are expressions that show the relationship where one expression is greater or less than another. In finding the domain of the logarithmic function \( y = \log(x^3 - x) \), we need to ensure the expression inside the log is positive: \( x^3 - x > 0 \).

Solving inequalities often involves manipulating the inequality just as you would an equation. However, special care must be taken so that the direction of the inequality remains valid, especially when multiplying or dividing by negative numbers. Here are the steps typically involved:

• Factor the Expression: Transform the expression into a product of factors, such as \( x(x-1)(x+1) \), which helps in identifying the critical points.
• Interval Testing: Use test points from each interval created by the critical points to determine if they satisfy the inequality.
• Draw on Number Line: Visual representation assists in identifying which sections solve your inequality.

By applying these principles, you can systematically determine the domain of a logarithmic function or similar expressions.

Factoring Expressions

Factoring is a powerful tool in algebra used to simplify and solve expressions and equations, especially when dealing with polynomials. For our function \( y = \log(x^3 - x) \), factoring the polynomial expression inside the log helps us solve the inequality \( x^3 - x > 0 \). Here's how it works:

• Factor Out Common Terms: Start by identifying and factoring out the greatest common factor, which in this problem is \( x \). This gives \( x(x^2 - 1) > 0 \).
• Recognize Special Patterns: The expression \( x^2 - 1 \) can further be factored using the difference of squares rule, resulting in \( x(x-1)(x+1) \). Factoring in this way exposes the roots easily, simplifying the process of solving inequalities.

Factoring reduces the complexity of handling large polynomial expressions. It transforms them into more manageable linear expressions. Once factored, it makes it easier to identify the zeros of the polynomial, thereby assisting in the analysis of the behavior of the function near these critical points.

Recognizing when to factor and how to use it effectively simplifies a variety of algebraic problems and is a key skill in finding the domain of functions.