Problem 66 - Precalculus

Question

State the domain of the logarithmic function in interval notation. $$f(x)=\log _{3}\left(x^{3}-3 x^{2}+3 x-1\right)$$

Step-by-Step Solution

Answer

The domain of the function is \((1, \infty)\).

Step 1: Identify the Argument of the Logarithm

The function is given as \( f(x) = \log_{3}(x^3 - 3x^2 + 3x - 1) \). The argument of the logarithm is \( x^3 - 3x^2 + 3x - 1 \).

Step 2: Determine When the Argument is Positive

The logarithm is only defined for positive arguments, so we need \( x^3 - 3x^2 + 3x - 1 > 0 \). This is an inequality we need to analyze and solve.

Step 3: Solve the Inequality

To determine the intervals in which the inequality holds, find the roots of the equation \( x^3 - 3x^2 + 3x - 1 = 0 \). Using the Rational Root Theorem or other methods like synthetic division, find that this factors as \( (x-1)^3 = 0 \), so \( x = 1 \) is a triple root. Evaluate the sign of \( x^3 - 3x^2 + 3x - 1 \) in intervals around \( x = 1 \).

Step 4: Determine the Domain

Test a number less than 1, such as \( x = 0 \), to see that \( 0^3 - 3\cdot0^2 + 3\cdot0 - 1 = -1 \) is less than zero. Thus, the function is negative when \( x < 1 \). If we test \( x = 2 \) (a number greater than 1), we find \( 2^3 - 3\cdot2^2 + 3\cdot2 - 1 = 1 \), which is positive. Therefore, the domain consists of numbers greater than 1, since the inequality is satisfied there. Thus the domain in interval notation is \( (1, \infty) \).

Key Concepts

Interval Notation: Defining the Domain of a Function

When we talk about interval notation, we are describing a way to represent a set of numbers, often the domain or range of a function. For the logarithmic function, it is crucial to determine where the argument (inside the logarithm) is positive, as the log is only valid for positive numbers. Once we've solved an inequality, the solutions indicate where the function operates correctly.
To describe these solutions concisely, we use interval notation:

• Use parentheses \((a, b)\) to denote all numbers between \(a\) and \(b\), not including \(a\) and \(b\) themselves.
• Use brackets \([a, b]\) to include \(a\) and \(b\) in the interval.
• Combine these symbols to express more complex domains, like \((a, b] \cup (c, d)\).

In the example, we found that the function \(f(x) = \log_3(x^3 - 3x^2 + 3x - 1)\) is defined for \(x > 1\). So, its domain is \((1, \infty)\), meaning all real numbers greater than 1.

Inequality Solving: Finding When Expressions Are Positive

Solving inequalities involves identifying when a variable expression is greater or less than a certain value. In our case, finding when the expression \(x^3 - 3x^2 + 3x - 1\) is positive is key to determining the domain.
First, solve the equation \(x^3 - 3x^2 + 3x - 1 = 0\) to find its roots. These roots help split the number line into intervals to test. Here, we discovered \(x = 1\) as a triple root. But this isn't enough — we need to test points outside and inside the intervals created by these roots.
For example, we tested \(x = 0\) and found the value was negative, and tested \(x = 2\) and found the value positive. These tests confirm that the function is positive for \(x > 1\). This method of solving inequalities by testing intervals is a common way to manage functions with complex polynomial expressions.

Rational Root Theorem: Finding Polynomial Roots

The Rational Root Theorem is a useful tool in finding the potential rational roots of a polynomial equation. It suggests that any rational solution, \(\frac{p}{q}\), to the polynomial equation must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient. This narrows down potential roots to test and can make finding the zeros easier.
However, in our scenario, we fully factor the polynomial \(x^3 - 3x^2 + 3x - 1\) to find \(x = 1\) as a triple root, mainly simplifying it to \((x-1)^3 = 0\). Knowing this tells us not only where the roots are, but that the behavior around these points is governed by this continuation of a single factor.
In cases where additional roots could exist, the Rational Root Theorem provides a systematic approach to guess and check potential roots, helping streamline analysis in polynomial equations.