Problem 77 - A Graphical Approach to Precalculus with Limits

Question

Determine the domain of each function. $$f(x)=\sqrt[3]{8 x-24}$$

Step-by-Step Solution

Answer

The domain of \( f(x) = \sqrt[3]{8x - 24} \) is all real numbers.

Step 1: Understand the Function

The function given is \( f(x) = \sqrt[3]{8x - 24} \). This is a cube root function, which means it takes the cube root of the expression \(8x - 24\).

Step 2: Identify Domain Restrictions

Cube root functions can accept any real number as input because both positive and negative numbers, as well as zero, have defined cube roots.

Step 3: Determine the Domain

Since cube root functions have no restrictions, the expression inside the cube root \(8x - 24\) can be any real number without causing undefined behavior. Thus, the domain of \(f(x)\) is all real numbers.

Key Concepts

Cube Root Function

A cube root function involves taking the cube root of a given expression. The cube root of a number \(x\) is a value \(y\) such that \(y^3 = x\). Unlike square roots, cube roots are defined for all real numbers, including negatives. This means that for any real number you pick, there is a real number result when taking the cube root.With a general form like \(g(x) = \sqrt[3]{x}\), these functions smoothly pass through the origin (0,0) on a graph. They rise and fall through negative and positive values symmetrically. Understanding this helps in recognizing why cube roots can handle all types of real inputs.

Real Numbers

Real numbers are an essential part of any mathematical framework. They include all the numbers you might think of in everyday mathematics: integers, fractions, and irrational numbers like \(\pi\) or \sqrt{2}. Real numbers fill the number line completely. They can be negative, positive, or zero.For functions, understanding that their domain can encompass all real numbers can make tackling problems easier. When a function’s domain includes all real numbers, it can handle any numerical substitution without causing undefined behavior. In the case of a cube root function, this means \(f(x) = \sqrt[3]{8x - 24}\) accepts any real number for \(x\).

Function Restrictions

Function restrictions arise when certain inputs can cause a function to behave unexpectedly, particularly becoming undefined. For example, square root functions are restricted because they cannot take the root of negative numbers without diving into complex numbers.However, cube root functions are unique in that they do not have these restrictions on real numbers. Whether \(8x - 24\) results in a positive, negative, or zero value, the cube root \(f(x)\) remains a real number. Therefore, unlike some functions, there are no traditional restrictions to watch out for in \(f(x) = \sqrt[3]{8x - 24}\). This characteristic is why its domain includes all real numbers. Knowing these potential restrictions, or lack thereof, can greatly influence how we solve and understand functions in mathematics.