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Chapter 10: Problem 100

Determine the domain of each function described. $$ f(t)=\sqrt[6]{4-3 t} $$

Short Answer

Expert verified
The domain of the function is \(-\infty < t \leq \frac{4}{3}\).

Step by step solution

01

Understand the Root Function

The given function is \[f(t)=\sqrt[6]{4-3t}\]This is a sixth root function. For even roots, the argument inside the root must be non-negative to avoid complex numbers.
02

Set the Argument to be Non-Negative

Ensure the expression inside the root, \[4-3t\], is greater than or equal to zero:\[4-3t \geq 0\]
03

Solve the Inequality

Solve for the variable t:\[4 \geq 3t\]Divide both sides by 3:\[\frac{4}{3} \geq t\] or\[t \leq \frac{4}{3}\]
04

Write the Domain

The domain of the function is all values of t that satisfy the inequality. Therefore, the domain is \[-\infty < t \leq \frac{4}{3}\].

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Key Concepts

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

root function
A root function involves extracting the root of a number or expression. In this exercise, we deal specifically with the sixth root. Understanding the nature of root functions is essential for determining their domain. For even roots like square roots, fourth roots, and sixth roots, the expression inside the root must be non-negative. This is because the root of a negative number results in a complex number. For example, \(\sqrt{16} = 4\), but \(\sqrt{-16}\) does not yield a real number. Similarly, in the given function \(f(t)=\sqrt[6]{4-3t}\), the number inside the root, \(4-3t\), has to be zero or positive to keep the values real.
inequality
In mathematics, inequalities like \(\geq\), \(>\), \(\leq\), and \(<\) help us compare values. To find the domain, we used an inequality to ensure non-negative values inside the root. For the function \(f(t)=\sqrt[6]{4-3t} \), we set up the inequality \(4-3t \geq 0\). By solving it, we determine the range of values for \(t\) that keeps the root function non-negative.

Here are key steps to solve the inequality:
  • Start with \(4-3t \geq 0\).
  • Rearrange to isolate \(t\).
  • Result: \(t \leq \frac{4}{3}\).
non-negative constraints
For any even root function to be defined over real numbers, the expression inside the root must be zero or positive. This is known as a non-negative constraint. In our function, \(4-3t\) must follow this rule. A non-negative constraint ensures that the function's output remains real.

To apply this to the domain, we set up an inequality \(4-3t \geq 0\), which means \(4 - 3t\) cannot be negative. Solving the inequality, we get \(t \leq \frac{4}{3}\).
This way, \(f(t)\) is a valid real number for all \(t\) satisfying \(t \leq \frac{4}{3}\).

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Most popular questions from this chapter

The absolute value of a complex number \(a+b i\) is its distance from the origin. Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}} .\) Find the absolute value of each complex number. $$ |-3-i| $$

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