Problem 77 - A Graphical Approach to College Algebra

Question

Determine the domain of each function. Do not use a calculator. $$f(x)=\sqrt[3]{8 x-24}$$

Step-by-Step Solution

Answer

The domain is all real numbers, \((-\infty, \infty)\).

Step 1: Understanding the Function Form

The given function is a cube root function, specifically \( f(x) = \sqrt[3]{8x - 24} \). For cube root functions, the expression inside the cube root does not have any restrictions from a domain perspective as compared to a square root function.

Step 2: Determine Domain for Cube Roots

Cube root functions like \( f(x) = \sqrt[3]{8x - 24} \) are defined for all real numbers because cube roots exist for all real values, whether positive, negative, or zero.

Step 3: Conclude the Domain

Since the expression inside the cube root has no restrictions and can take any real value, the domain of the function \( f(x) = \sqrt[3]{8x - 24} \) is all real numbers.

Key Concepts

Cube Root Function

The cube root function is a type of function that involves taking the cube root of an expression or number. It is expressed as \( \sqrt[3]{x} \), where \( x \) can be any real number. Unlike square root functions, cube root functions do not have restrictions on the value of \( x \).
This is because you can find the cube root of positive numbers, negative numbers, and even zero.

• For a positive \( x \), the cube root is positive.
• For a negative \( x \), the cube root is negative.
• For zero, the cube root is zero.

These properties make cube root functions very versatile in calculus and algebra since they are continuous and smooth over all real numbers. In graphical terms, they typically resemble a gentle "S" shape, going through the origin (0, 0), highlighting that they do not have asymptotes or breaks.

Real Numbers

Real numbers are the set of numbers that include both rational and irrational numbers. This set encompasses all numbers that we commonly use in everyday mathematics. As such, real numbers include:

• Whole numbers (like 0, 1, 2, 3...)
• Integers, both positive and negative (like -3, -2, -1, 0, 1, 2, 3...)
• Fractions and decimals (like 1/2, 0.75, etc.)
• Irrational numbers, which cannot be written as a simple fraction, like \( \pi \) or \( \sqrt{2} \)

Because cube root functions are defined for all real numbers, you can use any of these values as inputs (or \( x \) values) for the function.
This is distinct from some other functions, which may only allow certain values from this set based on their properties.

Function Domain

The domain of a function is the complete set of possible values of the independent variable that will produce a valid output from the function. In simpler terms, it’s about figuring out "what can go in" a function.
For the function \( f(x) = \sqrt[3]{8x - 24} \), the domain is all real numbers.
This is because, for cube root functions, there are no restrictions like there are for square root functions.
While square root functions require the expression under the root to be non-negative, cube root functions do not have this restriction. Hence, the expression \( 8x - 24 \) can be any real number, and the cube root of it will still yield a valid, real number value.Understanding the domain is crucial in functions because it tells us the scope within which the function operates without encountering any mathematical impossibilities or undefined operations.