Problem 19 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The domain of \( f(x)=\log(x^3-x) \) is \((-1, 0) \cup (1, \infty)\).

Step 1: Identify the Function Argument

The function given is \( f(x) = \log(x^3 - x) \). For the logarithmic function to be defined, its argument \( x^3 - x \) must be positive (greater than 0).

Step 2: Set the Inequality for the Argument

Set up the inequality \( x^3 - x > 0 \). This inequality will provide the domain of the function.

Step 3: Factor the Polynomial

The polynomial \( x^3 - x \) can be factored as \( x(x^2 - 1) \). This further simplifies to \( x(x - 1)(x + 1) \).

Step 4: Analyze the Factors

Each factor \( x \), \( (x - 1) \), and \( (x + 1) \) must be analyzed to determine when the product is positive. Examine the sign changes around the critical points \( x = 0 \), \( x = 1 \), and \( x = -1 \).

Step 5: Use a Sign Chart

Perform a sign analysis on the intervals created by the critical points: \( (-\infty, -1) \), \( (-1, 0) \), \( (0, 1) \), and \( (1, \infty) \).- For \( x < -1 \), all factors are negative, product is negative.- For \( -1 < x < 0 \), negative, positive, negative = positive.- For \( 0 < x < 1 \), positive, negative, positive = negative.- For \( x > 1 \), all factors are positive, product is positive.

Step 6: Identify Valid Intervals

Based on the sign chart, the product \( x(x-1)(x+1) \) is positive on the intervals \((-1, 0)\) and \((1, \infty)\).

Step 7: State the Domain

The domain of the function \( f(x) = \log(x^3 - x) \) is the union of intervals where \( x^3 - x > 0 \). Thus, the domain is \( (-1, 0) \cup (1, \infty) \).

Key Concepts

Logarithmic Function

A logarithmic function is a type of mathematical function that is closely related to exponential functions. Understanding logarithmic functions is crucial because they are used in many areas of mathematics and science, especially when dealing with growth and decay models or solving equations where the variable appears as the exponent. The basic idea behind a logarithmic function is finding the power, or exponent, that a base number must be raised to, to get another number. For example, in the logarithmic equation \( y = \log_b(x) \), \( b \) is the base, and \( x \) is the value for which we want to find the logarithm. This means that \( b^y = x \). Logarithms are the inverse of exponential functions.In the context of the function \( f(x)=\log(x^3-x) \), we need the argument \( x^3 - x \) to be positive because logarithms are only defined for positive numbers. This requirement forms the basis of finding the domain of a logarithmic function. By setting \( x^3 - x > 0 \), we ensure that the logarithmic function is defined everywhere within its domain.

Polynomial Inequality

Polynomial inequalities are mathematical expressions involving polynomials set with inequality symbols (such as \( >, <, \geq, \leq \)). Solving polynomial inequalities involves finding the values of the variable that make the inequality true. This is an important algebraic skill, especially when determining the domain of functions or solving real-world problems.To solve a polynomial inequality like \( x^3 - x > 0 \), we must determine where the expression is positive. This involves finding the roots of the polynomial, which simplifies the process of identifying intervals on the number line where the polynomial takes on positive or negative values. Essential steps include:

• Identifying the roots or critical points of the polynomial.
• Factoring the polynomial to make the problem more manageable.
• Using a "test-point" method or sign chart to determine the sign of the polynomial in different intervals.

When solving \( x^3 - x > 0 \), the challenge is to ensure the intervals where the polynomial is positive are correctly identified, which directly impacts the domain of functions like logarithms.

Factoring Polynomials

Factoring polynomials is a fundamental algebraic process that involves breaking down a polynomial into simpler "factor" polynomials whose product is the original polynomial. This technique is incredibly useful because it can simplify the process of finding roots and solving polynomial inequalities.For the polynomial \( x^3 - x \), the first step in factoring is recognizing common terms. Here, we can factor out \( x \) to get \( x(x^2 - 1) \). The expression \( x^2 - 1 \) is a difference of squares, which can be further factored into \( (x - 1)(x + 1) \).Thus, the complete factorization of \( x^3 - x \) is \( x(x - 1)(x + 1) \). This breakdown is essential because it helps to easily determine the sign changes of the polynomial by analyzing each factor individually in different intervals.

• For \( x(x - 1)(x + 1) \) to be positive, each factor's contribution to the product must be considered.
• Using a sign chart, we identify key points \( x = -1 \), \( x = 0 \), and \( x = 1 \), where the sign of the product could change.

Mastering the art of factoring is crucial for algebraic manipulation, solving equations, and understanding the behavior of polynomial functions.