Problem 81 - A Graphical Approach to College Algebra

Question

Determine the domain of each function. Do not use a calculator. $$f(x)=\sqrt{x^{3}-x}$$

Step-by-Step Solution

Answer

The domain of \( f(x) \) is \([-1,0] \cup [1, \infty) \).

Step 1: Understanding the Function

We begin by analyzing the function given: \( f(x) = \sqrt{x^3 - x} \). This function involves a square root, which dictates that the expression inside must be non-negative.

Step 2: Setting Up the Inequality

Since the square root function \( \sqrt{y} \) is only defined for \( y \geq 0 \), we need to ensure that \( x^3 - x \geq 0 \). This forms our inequality: \( x^3 - x \geq 0 \).

Step 3: Factoring the Expression

To solve \( x^3 - x \geq 0 \), we factor it as \( x(x^2 - 1) \geq 0 \). Further factoring gives \( x(x-1)(x+1) \geq 0 \).

Step 4: Finding Critical Points

The critical points, where the expression equals zero, are found by setting each factor equal to zero: \( x = 0 \), \( x = 1 \), and \( x = -1 \). These points divide the number line into intervals.

Step 5: Testing Intervals

We test intervals determined by the critical points: - For \( x < -1 \), pick \( x = -2 \): \((-2)((-2)-1)((-2)+1) < 0\).- For \( -1 < x < 0 \), pick \( x = -0.5 \): \( (-0.5)((-0.5)-1)((-0.5)+1) > 0\).- For \( 0 < x < 1 \), pick \( x = 0.5 \): \( 0.5(0.5-1)(0.5+1) < 0\).- For \( x > 1 \), pick \( x = 2 \): \( 2(2-1)(2+1) > 0\).

Step 6: Constructing the Domain

From testing intervals and including critical points where the expression equals zero, we find that the function is defined for intervals \( x \in [-1,0] \cup [1, \infty) \). Therefore, the domain is these combined intervals.

Key Concepts

Critical points

When analyzing the domain of a function involving inequalities, identifying critical points is a key step. These critical points are where the expression equals zero or becomes undefined, and they help us determine the intervals to test. For the function \( f(x) = \sqrt{x^3 - x} \), setting the equation inside the square root to zero reveals these points.To solve \( x(x-1)(x+1) = 0 \), we find:

• \( x = 0 \)
• \( x = 1 \)
• \( x = -1 \)

These points divide the real number line into distinct intervals, which are crucial for determining where the function is defined. By testing these intervals, we can subsequently pin down the domain of the function. Identifying critical points is like setting the stage for testing the intervals you derive.

Inequalities

Understanding inequalities is vital when working with functions defined by square roots, like \( f(x) = \sqrt{x^3 - x} \). Since the square root function is defined only for non-negative numbers, we need to set up an inequality to find solutions where the function is defined.Here, we establish:\[ x^3 - x \geq 0 \]Solving this inequality involves ensuring each part of the factored expression \( x(x-1)(x+1) \geq 0 \) meets the condition across different intervals. Inequalities guide us to identify where and how our function behaves throughout the number line. They link the mathematical properties of the expression to the concept of domain, underpinning how we frame those behaviors mathematically.

Factoring

Factoring simplifies solving inequalities, especially when determining domains of complex functions. For \( f(x) = \sqrt{x^3 - x} \), factoring \( x^3 - x \) is an essential step.Initially, we notice a common factor:

• Factor out \( x \): \( x(x^2 - 1) \)
• Recognize the difference of squares in \( x^2 - 1 \): \( x(x-1)(x+1) \)

By breaking down this expression, we create a clearer path to solve the inequality \( x(x-1)(x+1) \geq 0 \). Each factor individually represents a component that, when considered collectively, dictates how we test intervals. Factoring is like untangling a puzzle, making complex expressions more manageable, and revealing key critical points.

Interval testing

Interval testing helps determine which parts of the real number line make an inequality true, and thus where our function is defined. Once critical points are identified, these points are used to divide the number line into segments or intervals.In the context of \( f(x) = \sqrt{x^3 - x} \), we test intervals created by critical points \( x = -1, 0, \) and \( 1 \):

• Select a value from each interval: \(-2, -0.5, 0.5, 2\)
• Plug these into the factored inequality \( x(x-1)(x+1) \)
• Determine if the result satisfies \( \geq 0 \)

For example:

• \(-2((-2)-1)((-2)+1) < 0\)
• \(-0.5((-0.5)-1)((-0.5)+1) > 0\)

Conducting interval testing provides insight into which intervals satisfy the inequality and hence define the function's domain. It’s a systematic way to combine critical points and inequalities for a comprehensive solution.