Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the domain of the function. $$f(x)=\sqrt[4]{x^{2}+3 x}$$

Step-by-Step Solution

Answer

The domain for the function \(f(x)=\sqrt[4]{x^{2}+3 x}\) is \([-3, 0] \cup [0, +\infty)\), or more simply, \([-3, +\infty)\).

Step 1: Setting up the inequality

For the 4th root to be real and defined, the value under the root must be greater than or equal to zero. Therefore, we need to solve the inequality \(x^{2}+3x \geq 0\).

Step 2: Factor and solve the inequality

Let's first factor the expression. We can write \(x^{2}+3x\geq 0\) as \(x(x + 3) \geq 0\). Now, the solution to the inequality will come from where each factor is equal to zero or switches sign. This happens at \(x = 0\) and \(x = -3\).

Step 3: Test intervals and find the domain

To find the intervals where the expression is positive or zero, pick a test point from each of the intervals (-infinity, -3), (-3, 0) and (0, infinity), and substitute into the expression \(x(x + 3)\). If the result is positive or zero, that interval is part of the solution. Testing gives: Interval (-infinity, -3): Choose \(x = -4\): \(-4*(-4 + 3) = -4\), which is less than 0. So, this interval is not in the domain. Interval (-3, 0): Choose \(x = -1\): \(-1*(-1 + 3) = 2\), which is greater than 0. So, this interval is in the domain. Interval (0, infinity): Choose \(x = 1\): \(1*(1 + 3) = 4\), which is greater than 0. So, this interval is also in the domain.

Key Concepts

Inequality Solving

Solving inequalities is essential when determining regions where expressions are valid. In this exercise, we explore the inequality \(x^2 + 3x \geq 0\) to find where the fourth root function is defined. Inequality solving involves identifying values of \(x\) which satisfy an inequality. The steps include:

• Rephrasing the original condition into an inequality form that needs solving.
• Factoring or other algebraic manipulations to get simpler expressions.
• Determining the values of \(x\) where these simplified expressions keep the entire inequality true.

After simplifying and factoring, you can determine the intervals that satisfy the given inequality by testing different sections within the possible values for \(x\). This practice is integral to revealing where the function maintains its defined nature for real outputs.

Real Numbers

Real numbers include all the numbers you could think of on the number line, from minus infinity to plus infinity. They encompass everything: integers, fractions, and irrational numbers like \(\sqrt{2}\).

In solving inequalities like \(x^2 + 3x \geq 0\), you're aiming to find which real numbers make the expression true. This underscores the necessity of understanding the continuum of the number line. For example, when determining the domain of the function \(f(x) = \sqrt[4]{x^2 + 3x}\), only certain real number segments make the expression logically valid. Therefore, understanding these segments demands knowledge of how real numbers interact with algebraic expressions and inequalities.

Interval Testing

Interval testing is a practical approach to identifying which parts of the number line meet certain conditions. When solving inequalities such as \(x(x + 3) \geq 0\), breaking it down into intervals can be valuable.First, determine critical points, where either the expression is zero or changes sign. Here, critical points were found at \(x = 0\) and \(x = -3\). The next step is to test values in the resulting intervals: (-∞, -3), (-3, 0), and (0, ∞).

• Pick any value within each interval and substitute it into the inequality.
• If the expression is positive (or zero), the interval is part of the solution.
• If the expression is negative, the interval is not part of the domain.

By rigorous testing, you can confidently say which intervals should be included in the function's domain to ensure the expression is defined and valid in its real number scope.

Factoring Expressions

Factoring is a fundamental algebraic tool to simplify expressions or solve equations and inequalities. For the inequality \(x^2 + 3x \geq 0\), we factor it as \(x(x + 3)\). This helps in breaking down the expression into simpler components and identifying roots or critical values.

When an expression is factored:

• You can quickly identify key points or roots by setting each factor to zero.
• Factoring simplifies the process of finding intervals needed for testing or solving.
• It helps determine the overall behavior of the function across different ranges of x-values.

In this case, the critical points \(x = 0\) and \(x = -3\) are clearly visible once the quadratic expression is factored, making subsequent steps like interval testing, straightforward. Factoring provides clarity and reduces complexity, aiding in accurately finding where the entire inequality holds true.