Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Determine the domain of each function. $$f(x)=-\sqrt[4]{6-x}$$

Step-by-Step Solution

Answer

The domain is all real numbers \( x \) such that \( x \leq 6 \).

Step 1: Understanding the Function

The function given is \( f(x) = -\sqrt[4]{6-x} \). It involves a fourth root, which requires its argument to be non-negative to be defined for real numbers.

Step 2: Setting the Argument of the Root

The expression inside the fourth root, \( 6-x \), must be greater than or equal to 0. This is because the fourth root, like any even root, is only defined for non-negative numbers.

Step 3: Solving the Inequality

Set up the inequality: \( 6-x \geq 0 \). This can be rearranged to \( x \leq 6 \).

Step 4: Conclusion of the Domain

Since \( x \) can be any real number that makes \( 6-x \) non-negative, the domain of the function \( f(x) \) is all real numbers \( x \) such that \( x \leq 6 \).

Key Concepts

Fourth Root

The fourth root of a number is a special type of mathematical operation. If you have a number, say 16, and you want to find its fourth root, you're essentially looking for a number that, when multiplied by itself four times, equals 16.
This is denoted as \(\sqrt[4]{16} = 2 \)% because \(2 \times 2 \times 2 \times 2 = 16 \). When dealing with even roots, such as square roots or fourth roots, the number inside the root must be non-negative if we want the result to be a real number.

• Even roots are always taken of non-negative numbers when working within the realm of real numbers.
• The logic behind this is simple: multiplying a negative number by itself an even number of times yields a positive number.

When solving problems involving fourth roots, remember that the argument under the root sign must be equal to or greater than zero for real and meaningful answers.

Real Numbers

Real numbers comprise a broad range of values you frequently encounter in math. They include numbers that you count with (integers), fractions (rational numbers), and numbers that can't be expressed as fractions (irrational numbers).
In general, real numbers can be thought of as any point along the number line. This includes positive numbers, negative numbers, and even zero.

• Integers are numbers without fractions. Examples include -3, 0, 7.
• Rational numbers can be expressed as fractions, like \(\frac{3}{4}\) or 0.75.
• Irrational numbers are not expressible as fractions and often include constants like \(\pi\) and \(\sqrt{2}\).

In problems involving functions, especially with roots, the domain of a function is usually specified in terms of real numbers. If a function like the one in the original exercise is to be defined and useful, its inputs must align with real number properties.

Inequality

Inequalities are expressions that show the relationship between two values when they are not equal. They are used in mathematics to describe the range of possible values of a variable. An inequality could be as simple as \(x > 3\), or more complex, like \(6 - x \geq 0\).
Inequalities help specify the domain of many math functions.

• The symbol \(<, >, \leq, \geq\) are commonly used in inequalities.
• There are a few fundamental rules when working with them, such as flipping the inequality sign when multiplying or dividing by a negative number.

In the original exercise, the inequality \(6 - x \geq 0\) is used to determine when the fourth root of \(6-x\) is a real number.
Solving this inequality helps identify which numbers are valid inputs (domain) for the function. The solution of \(6 - x \geq 0\) gives us \(x \leq 6\), which is crucial for understanding the function's behavior.