Problem 26 - College Algebra with Corequisite Support

Question

Find the domain of the function \(f(x)=\sqrt{2 x^{3}-50 x}\) by: (a) using algebra. (b) graphing the function in the radicand and determining intervals on the \(x\) -axis for which the radicand is nonnegative.

Step-by-Step Solution

Answer

Domain: \([-5, 0] \cup [5, \infty)\).

Step 1: Set Up the Inequality

The function \(f(x) = \sqrt{2x^3 - 50x}\) is defined only when the expression inside the square root (the radicand), \(2x^3 - 50x\), is nonnegative. Thus, we solve for \(2x^3 - 50x \geq 0\).

Step 2: Factor the Inequality

The expression \(2x^3 - 50x\) can be factored by extracting the greatest common factor, which is \(2x\). Thus, we rewrite it as \(2x(x^2 - 25)\). This quadratic expression can further be factored as \(2x(x - 5)(x + 5)\).

Step 3: Finding Critical Points

Set each factor equal to zero to find the critical points: \(2x = 0\), \(x - 5 = 0\), and \(x + 5 = 0\). This yields the critical points \(x = 0\), \(x = 5\), and \(x = -5\).

Step 4: Test the Intervals

These critical points divide the real number line into the intervals: \((-\infty, -5)\), \((-5, 0)\), \((0, 5)\), and \((5, \infty)\). Test each interval in the inequality \(2x(x-5)(x+5) \geq 0\) to determine where it holds true. For instance:- Pick \(x = -6\) in \((-\infty, -5)\); \(2x(x-5)(x+5) < 0\).- Pick \(x = -2\) in \((-5, 0)\); \(2x(x-5)(x+5) > 0\).- Pick \(x = 2\) in \((0, 5)\); \(2x(x-5)(x+5) < 0\).- Pick \(x = 6\) in \((5, \infty)\); \(2x(x-5)(x+5) > 0\).

Step 5: Compile the Domain from Valid Intervals

The function is defined where the inequality is satisfied, which is in the intervals \((-5, 0)\cup (5, \infty)\), including the points \(-5\), \(0\), and \(5\) themselves since the inequality signs are non-strict (\(\geq\)). Thus, the domain is \([-5, 0]\cup [5, \infty)\).

Step 6: Graphical Verification

Sketch the graph of the radicand function \(y = 2x^3 - 50x\). Identify where the graph is on or above the x-axis, which corresponds to nonnegative values. It crosses the x-axis at the points \(x = -5\), \(x = 0\), and \(x = 5\), confirming the intervals \([-5, 0]\) and \([5, \infty)\).

Key Concepts

Algebraic Inequality

When dealing with algebraic inequalities, it means we're seeking to understand where an algebraic expression is greater than or equal to zero. In this exercise, the function is defined based on where its radicand, which is inside the square root, is nonnegative. This is because, in mathematical principles, a square root of a negative number is not defined in the realm of real numbers. Therefore, the inequality is set up as follows:

• Start with the inequality: \(2x^3 - 50x \geq 0\).
• This establishes the regions where the function's square root is well-defined.

By solving this inequality, we can determine the domain where the function naturally exists.

Critical Points

Critical points are specific values of \(x\) at which the value of the expression changes its nature, often due to its factors equaling zero. Here, finding critical points involves setting each factor of the function's expression to zero. This step is essential because these points can signal changes in the positivity or negativity of the entire expression.

• For the factor \(2x\), the critical point is \(x = 0\).
• For \(x - 5\), solve \(x - 5 = 0\) to find \(x = 5\).
• Similarly, \(x + 5 = 0\) yields \(x = -5\).

These points divide the expression's behavior into different intervals on the number line, which can then be tested for their validity in the inequality.

Factorization

Factorization simplifies equations and inequalities by breaking them down into products of simpler expressions. This technique allows us to see more clearly the zeros of the function and, importantly, to set up interval testing with its critical points. For the radicand \(2x^3 -50x\), factorization proceeds as follows:

• First, extract the greatest common factor, which is \(2x\).
• This gives \(2x(x^2 - 25)\).
• Recognizing the remaining expression as a difference of squares, \(x^2 - 25\) can be further written as \((x - 5)(x + 5)\).

Ultimately, the expression factors into \(2x(x - 5)(x + 5)\), facilitating further analysis through critical point and interval testing.

Interval Testing

Interval testing is a systematic way to determine on which intervals the inequality holds true. Once the critical points are identified, they divide the number line into segments that can be analyzed separately. Here's the step-by-step guide for performing interval testing:

• Divide the number line into intervals based on the critical points: \((-\infty, -5)\), \((-5, 0)\), \((0, 5)\), and \((5, \infty)\).
• Pick a test point from each interval. For instance, choose \(x = -6\) in the interval \((-\infty, -5)\).
• Substitute this test point into the expression \(2x(x-5)(x+5)\).
• Determine the sign of the resulting value. If positive, the inequality holds for that interval.

After testing, it's revealed that the intervals \((-5, 0)\) and \((5, \infty)\) satisfy the inequality, leading us to conclude the domain of the function. This smart, step-by-step division and testing is how mathematicians simplify complex expressions.