Problem 76 - A Graphical Approach to College Algebra

Question

Determine the domain of each function. Do not use a calculator. $$f(x)=-\sqrt[4]{2-0.5 x}$$

Step-by-Step Solution

Answer

The domain is \((-\infty, 4]\).

Step 1: Identify Restrictions

Since the function involves a fourth root, the expression under the root, \(2 - 0.5x\), must be non-negative. The restriction comes from the fact that the root of a negative number is not defined in the set of real numbers.

Step 2: Solve the Inequality

Set up the inequality that the expression \(2 - 0.5x\) must be greater than or equal to zero: \(2 - 0.5x \geq 0\).

Step 3: Rearrange the Inequality

Subtract 2 from both sides of the inequality to start solving it: \(-0.5x \geq -2\).

Step 4: Solve for x

Divide both sides of the inequality by \(-0.5\). Remember to flip the inequality sign since you are dividing by a negative number: \(x \leq 4\).

Step 5: Write the Domain

Since \(x\) can be any real number less than or equal to 4, the domain of the function is \( (-\infty, 4] \).

Key Concepts

Understanding Inequalities in Algebra

When determining the domain of a function, especially one that involves roots or radicals, inequalities help us find out where the function is defined. For the given function,

• We identified that the expression inside the fourth root, \(2 - 0.5x\), must be greater than or equal to zero. This is because taking roots of negative numbers is not allowed in real numbers.
• Setting up an inequality, we began with \(2 - 0.5x \geq 0\).
• From there, solving the inequality in steps allowed us to isolate \(x\).
• Remember to consider reversing the inequality when dividing by a negative number.

Algebraic manipulation of inequalities is crucial in finding valid ranges for variables to ensure the function produces real, defined outputs.

Exploring Real Numbers

The set of real numbers includes all numbers on the number line. They encompass integers, fractions, and irrational numbers like \(\pi\) and \(\sqrt{2}\). Real numbers are essential in algebra because:

• They form the basis for understanding domain restrictions.
• Function outputs that aren’t real numbers (like imaginary numbers) are typically outside the domain we consider.

In the function \(-\sqrt[4]{2 - 0.5x}\), we needed \(2 - 0.5x\) to be non-negative, ensuring the result is real. Thus, solving for when the expression is \(\geq 0\) provides a valid set of \(x\) values, keeping the outputs real.

Roots and Radicals in Functions

Roots and radicals, such as square roots or fourth roots, dictate important domain restrictions. The general rule:

• The expression inside a root must be non-negative for real outputs.
• For example, \(\sqrt[4]{2 - 0.5x}\) must have a non-negative inside expression to be defined in real numbers.
• This led us to the inequality \(2 - 0.5x \geq 0\).

Working through radicals requires careful handling of inequalities to ensure solutions make sense. Missteps in solving can lead to interpreting non-real results, which are not suitable for real number domains.