Problem 13 - College Algebra

Question

For the following exercises, find the domain of each function using interval notation. $$ f(x)=\sqrt[3]{x-1} $$

Step-by-Step Solution

Answer

The domain of \( f(x) = \sqrt[3]{x-1} \) is \( (-\infty, \infty) \).

Step 1: Understand the Function

The function given is \( f(x) = \sqrt[3]{x-1} \). This is a cube root function of the expression \( x-1 \).

Step 2: Identify the Domain for Cube Root Functions

Cube root functions are defined for all real numbers. This is because the cube root of any real number (positive, negative, or zero) is also a real number. Hence, there are no restrictions on \( x \) for \( f(x) = \sqrt[3]{x-1} \).

Step 3: Express the Domain in Interval Notation

Since the function is defined for all real numbers, the domain is the entire set of real numbers. In interval notation, this is expressed as \( (-\infty, \infty) \).

Key Concepts

Interval Notation

Interval notation is a concise way of describing sets of numbers, typically representing the domain or range of a function. It uses brackets and parentheses to indicate the inclusivity or exclusivity of the boundary values.

• "(" and ")" are used for open intervals, meaning the end values are not included.
• "[" and "]" signify closed intervals, where the end values are included.

For example, the interval \((-3, 5]\) means that the set includes all real numbers greater than -3 up to and including 5. Using interval notation makes it easier to express the domain of a function, especially when it includes a large set of numbers. This notation helps to simplify complex concepts into straightforward expressions.

Cube Root Functions

Cube root functions involve the cube root of a variable or expression, represented as \( f(x) = \sqrt[3]{x}.\) Unlike square roots, cube roots can be computed for all real numbers, including negative ones. This is because cubing a negative number results in a negative number, but cube rooting one returns it to its original state.

• Domain of Cube Root Functions: Since the cube root of any real number yields a real result, the domain of a cube root function is all real numbers.

For example, given \( f(x) = \sqrt[3]{x-1}\), the expression within the cube root can be any real number. Thus, there are no restrictions on the values \(x\) can take.This makes cube root functions very flexible in mathematical operations.

Real Numbers

Real numbers include all the numbers along the number line. They encompass rational numbers (like integers and fractions) as well as irrational numbers (numbers which cannot be written as simple fractions, such as \( \pi \) and \( \sqrt{2}\)). Real numbers are used to express quantities that have magnitude and placement along a number line in mathematics.
Here are some key subcategories of real numbers:

• Rational Numbers: Numbers that can be divided into two integers, expressed as a fraction.
• Irrational Numbers: Non-repeating, non-terminating decimals.

Real numbers are fundamental in defining domains of functions because they represent every possible value \(x\) can assume for a function like \( f(x) = \sqrt[3]{x-1}\). This includes zero, positive, and negative numbers.

Function Domain Definitions

The domain of a function is a critical concept in mathematics. It represents all the possible input values (or \(x\)-values) for which the function is defined. Understanding domain allows us to know what can be "plugged into" the function without causing mathematical inconsistencies.
There are a few important things to consider when determining the domain:

• Denominators: In rational functions, any value that makes a denominator zero is excluded.
• Radicals: Even-index roots cannot have negative values inside the radical.
• Logarithms: Cannot have zero or negative arguments.

For the function \( f(x) = \sqrt[3]{x-1}\), the only consideration is whether cube roots are universally defined, which they are for all real numbers. Thus, the domain is unrestricted: \(x\) can be any real number, specified concisely in interval notation as \((-\infty, \infty)\).