Problem 6 - Elementary and Intermediate Algebra

Question

Select the appropriate word to complete each of the following. The domain of the function \(f\) given by \(f(x)=\sqrt[3]{x}\) is the set of all ______ numbers..

Step-by-Step Solution

Answer

real

Step 1: Understand the function

The given function is a cube root function, represented as \( f(x) = \sqrt[3]{x} \). A cube root is defined for all real numbers, unlike a square root which is only defined for non-negative numbers.

Step 2: Analyze the domain

The domain of a function is the set of all possible input values (x-values) that the function can accept. For the cube root function, there are no restrictions on the values of \( x \).

Step 3: Conclude the domain

Since the cube root function can accept any real number as an input, the domain of \( f(x) = \sqrt[3]{x} \) is all real numbers.

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics and include all the numbers you can think of that are not imaginary. They consist of rational numbers, such as fractions and integers, and irrational numbers, which cannot be expressed as fractions. This includes numbers like 3, -1, 0.5, and \(\frac{22}{7}\), as well as numbers like \( \pi \) and \( \sqrt{2} \).

Real numbers can be positive, negative, or zero. They are used in all sorts of everyday calculations and play a crucial role in various mathematical functions, including the cube root function. For any function, especially when you are identifying its domain, understanding that all real numbers include every possible number you can place on a number line, except complex or imaginary numbers, is essential.

Function Domain

The domain of a function is the set of all possible input values that the function can accept. Simply put, it's 'where the function lives.' For instance, the function \( f(x) = x^2 \) is defined for all real numbers since you can square any real number.

However, certain functions have restrictions. For example, the square root function \( f(x) = \sqrt{x} \) is only defined for non-negative numbers because you can't take the square root of a negative number and get a real result. Understanding the domain helps determine the values that make the function work without errors or undefined behaviors.

Cube Root

A cube root function is a specific type of mathematical function. It is written as \( f(x) = \sqrt[3]{x} \) and is defined for all real numbers. Unlike the square root function, which is restricted to non-negative numbers, the cube root function can take both positive and negative values.

\r This function operates with the principle that the cube root of a number \( y \) is a number \( x \) such that \( x^3 = y \). For instance, the cube root of 27 is 3 because \( 3^3 = 27 \), and the cube root of -27 is -3 because \( (-3)^3 = -27 \).

\r< The key understanding here is that the cube root function \( f(x) = \sqrt[3]{x} \) can handle any real input, making its domain the entire set of real numbers. This inclusivity differentiates it from functions like the square root and allows for a broader range of applications.