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Chapter 10: Problem 99

Determine the domain of each function described. $$ f(t)=\sqrt[5]{8-3 t} $$

Short Answer

Expert verified
The domain of \( f(t) = \sqrt[5]{8 - 3t} \) is all real numbers.

Step by step solution

01

Identify the Function Type

The given function is \( f(t) = \sqrt[5]{8 - 3t} \). Notice that this is a fifth root function.
02

Understand the Domain of a Root Function

For any odd root function, such as a cube root or fifth root, the radicand (the expression inside the root) can take any real number value. There are no restrictions.
03

State the Domain

Therefore, the domain of \( f(t) = \sqrt[5]{8 - 3t} \) is all real numbers.
04

Express the Domain in Set Notation

The domain can be written in set notation as \( \{ t | t \in \mathbb{R} \} \), where \(\mathbb{R} \) represents the set of all real numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fifth Root Function
Understanding the fifth root function is crucial to solve problems like this. The fifth root function is represented mathematically as \( f(x) = \sqrt[5]{x} \). This notation means we're looking for a number which, when multiplied by itself five times, yields \( x \). Unlike square roots, fifth roots can be taken from both positive and negative numbers, as raising a negative number to an odd power results in a negative number. Substituting \( x \) with a variable expression doesn鈥檛 change this property.
Radicand
Each root function has an expression inside the root symbol known as the radicand. For the given function \( f(t) = \sqrt[5]{8 - 3t} \), the radicand is \( 8 - 3t \). Comprehending how the radicand influences the domain of the function is key. Since we're dealing with an odd root here, the radicand \( 8 - 3t \) can be any real number. There are no specific restrictions on the value of the radicand in the case of odd roots.
Set Notation
After determining the domain of a function, it's useful to express it clearly in set notation. For the function \( f(t) = \sqrt[5]{8 - 3t} \), the domain is all real numbers because the radicand can take any real value. We express this in set notation as \{ t | t \( \in \mathbb{R} \) \}. This notation reads as: the set of all \( t \), such that \( t \) is an element of the real numbers. It succinctly captures that any real value for \( t \) will result in a valid output for the function.

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