Problem 25 - Calculus with Concepts in Calculus

Question

Find the domain of the function. $$ f(t)=\sqrt[3]{1-t^{2}} $$

Step-by-Step Solution

Answer

The domain of \( f(t) = \sqrt[3]{1-t^2} \) is all real numbers \(( -\infty, +\infty )\).

Step 1: Understand the Function Type

The function is given by \( f(t) = \sqrt[3]{1-t^2} \). This involves a cube root, and remember that cube roots are defined for all real numbers.

Step 2: Determine Restrictions

For cube roots like \( \sqrt[3]{1-t^2} \), there are no restrictions on \( t \) when considering real numbers. This is because any real number can have a real cube root, unlike square roots which require non-negative radicands.

Step 3: State the Domain

Since there are no restrictions due to the cube root, the domain is the set of all real numbers. Therefore, \( t \) can be any real number.

Key Concepts

Domain of a Function

The domain of a function refers to the complete set of possible values of the independent variable, often represented by symbols such as x or t in function notation. Put simply, it's the "input" values that a function can accept without encountering mathematical errors.

For many functions, you have to be careful about which values you can plug in. For instance:

• In rational functions, you should never have the denominator equal to zero.
• For square root functions, ensure the expression under the root is non-negative.

In the exercise, the function is given by \( f(t) = \sqrt[3]{1-t^2} \). This falls under the category of cube root functions. Thanks to the inherent properties of cube roots, we don't have as many restrictions as in other types.

Rather wonderfully, this means that for the function \( f(t) \), there are no restrictions on \( t \). Any real number can be used as an input, making the domain the set of all real numbers. Understanding the domain helps students predict function behavior and solve equations tailored to specific domains.

Cube Root

The cube root of a number refers to a value that, when multiplied by itself three times, gives the original number. In symbolic terms, if \( x = \sqrt[3]{a} \), then \( x^3 = a \). Cube roots are indicated by the symbol \( \sqrt[3]{\cdot} \).

A distinct feature of cube roots is that they are defined for all real numbers, both positive and negative. This is unlike square roots, which are only defined for non-negative numbers:

• The cube root of a positive number is positive.
• The cube root of zero is, quite plainly, zero.
• The cube root of a negative number is negative.

This ability to take cube roots of negative numbers arises from how multiplying three negative numbers together still gives a negative result.

For example, \( \sqrt[3]{-8} = -2 \) since \((-2) \times (-2) \times (-2) = -8\). This property plays a critical role in widening the domain of functions like the one in the original exercise, because there are no restrictions on the input values.

Real Numbers

Real numbers are the collection of all numbers that can be found on the number line. They include:

• Positive numbers like 2, 3.5, and \( \sqrt{16} \).
• Negative numbers such as -1 and -\( \pi \).
• Zero.
• All rational numbers, which are numbers that can be expressed as the fraction of two integers, like \(\frac{3}{4}\).
• All irrational numbers, which cannot be expressed as simple fractions, like \(\pi\) and \(\sqrt{2}\).

Real numbers are essential in mathematics because they form a complete system, meaning that they can be used in accounting for almost all typical calculations and equations in everyday life. In function analysis, understanding the set of real numbers ensures you appreciate its infinite nature and inclusivity. This is crucial for comprehending why certain functions, like those involving cube roots, have the domain of all real numbers.