Problem 47 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Domain is \((-\infty, +\infty)\).

Step 1: Identify Function Type

The function given is a cube root function, \( f(t) = \sqrt[3]{t-1} \). Cube root functions differ from square root functions in terms of domain because cube roots can be taken for any real number.

Step 2: Determine Domain for Cube Root Function

Cube root functions, unlike square root functions, are well-defined for all real numbers. This is because the cube root of a negative number or zero is also a real number. Therefore, the expression \( t-1 \) can take any real number value.

Step 3: Domain Conclusion

Since \( f(t) = \sqrt[3]{t-1} \) is defined for all real values of \( t \), the domain of the function \( f(t) \) is all real numbers, represented as \( (-\infty, +\infty) \).

Key Concepts

Cube Root Function

The cube root function is a unique type of mathematical function. It involves finding a number which, when used three times in a multiplication (multiplied by itself twice), results in the given number. The cube root of a number \( x \) is denoted by \( \sqrt[3]{x} \). For instance, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \). Similarly, the cube root of -8 is -2, as \( (-2) \times (-2) \times (-2) = -8 \).
Cube root functions, like \( f(t) = \sqrt[3]{t-1} \), are particularly straightforward in terms of their domain, because cube roots are defined for all real numbers. In contrast to square root functions, where you cannot have a negative number inside the root, cube roots can handle both positive and negative numbers. This is because raising a negative number to an odd power results in a negative number, making cube root of negative numbers manageable. Therefore, there are no restrictions on values \( t \) can take for a cube root function.

Real Numbers

Real numbers are the most extensive and familiar set of numbers used in mathematics. They include both rational numbers, like integers and fractions, and irrational numbers that cannot be expressed as simple fractions. Some examples of irrational numbers include \( \pi \) and \( \sqrt{2} \).
Real numbers are represented on a continuous number line that extends indefinitely in both positive and negative directions. This comprehensive set includes:

• Natural numbers: 1, 2, 3, ...
• Whole numbers: 0, 1, 2, 3, ...
• Integers: ..., -3, -2, -1, 0, 1, 2, ...
• Rational numbers: numbers that can be expressed as fractions, such as \(\frac{1}{2}\) and \(\frac{5}{7}\)
• Irrational numbers: numbers that cannot be written as fractions, such as \( \sqrt{2} \) and \( \pi \)

In our exercise involving the cube root function \( f(t) = \sqrt[3]{t-1} \), the variable \( t \) can take any real number, meaning \( t \) can be any value on this vast number line.

Function Domain

Understanding the domain of a function is crucial when working with functions in mathematics. The domain of a function is the complete set of possible input values (often denoted as \( x \) or \( t \)) for which the function is defined and produces a real number output.
For cube root functions like \( f(t) = \sqrt[3]{t-1} \), the domain is particularly broad. Because the cube root is capable of handling any real number—whether positive, negative, or zero—the expression \( t-1 \) can be any value along the real number line. This feature means there are no restrictions on \( t \), making the domain of \( f(t) \) encompass all real numbers.
To express this, mathematicians use interval notation. The domain of \( f(t) = \sqrt[3]{t-1} \) is written as \((-\infty, +\infty)\). This notation signifies that \( t \) can be any number starting from negative infinity to positive infinity.