Problem 53 - Algebra and Trigonometry

Question

Find the domain of the function. $$ f(t)=\sqrt[3]{t-1} $$

Step-by-Step Solution

Answer

The domain of \( f(t) = \sqrt[3]{t-1} \) is all real numbers, \( t \in \mathbb{R} \).

Step 1: Understanding the Problem

We need to find the domain of the function \( f(t) = \sqrt[3]{t-1} \). The domain of a function consists of all the possible input values (in this case, values of \( t \)) that keep the function defined.

Step 2: Identifying the Function Type

The given function involves a cube root: \( f(t) = \sqrt[3]{t-1} \). Cube roots, unlike square roots, are defined for all real numbers.

Step 3: Determining the Domain Constraints

For a cube root function \( \sqrt[3]{x} \), there are no restrictions on \( x \). Therefore, \( t - 1 \) can be any real number.

Step 4: Expressing the Domain

Since \( t - 1 \) can be any real number, \( t \) can also be any real number. Thus, the domain of the function is all real numbers.

Key Concepts

Cube Root Function

A cube root function is a type of mathematical function that involves finding the cube root of an expression. In this case, we are looking at \( f(t) = \sqrt[3]{t-1} \).

Cube roots are unique compared to square roots because they are defined for both positive and negative values. This means:

• There is no restriction on the values you plug into the function, as cube roots can handle any real number input.
• The operation makes it possible to calculate the cube root of negative numbers too.

For example, the cube root of \(-8\) is \(-2\) because \((-2) \times (-2) \times (-2) = -8\). This property highlights why the domain of a cube root function is often all real numbers.

Real Numbers

Real numbers include all the numbers you can think of on the number line: both positive and negative numbers, including zero, and every number in between. Real numbers can be classified into various categories, such as:

• Integers: Whole numbers that include positive numbers, zero, and negative numbers without fractions or decimals.
• Rational numbers: Numbers that can be expressed as the quotient of two integers (fractions, including repeating or terminating decimals).
• Irrational numbers: Numbers that cannot be expressed exactly as fractions, such as \( \pi \) or \( \sqrt{2} \).

Understanding that cube root functions accept all real numbers means realizing that no number is excluded from the domain. This inherent flexibility of real numbers is what allows functions like \( f(t) = \sqrt[3]{t-1} \) to have a domain of all real numbers.

Function Domain Constraints

Domain constraints refer to limitations on what can be input into a function to produce a valid output. In some functions, you might need to restrict the domain to avoid mathematical errors like division by zero or taking the square root of a negative number.

For cube root functions, such as \( f(t) = \sqrt[3]{t-1} \), these typical constraints don't apply because:

• The cube root of any real number is still a real number.
• There are no values of \( t \) that result in an undefined or invalid expression under the cube root.

Consequently, when assessing the domain of cube root functions, you simply state that all real numbers are included, confirming that there are no restrictions required to keep the function defined.