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Chapter 7: Problem 101

Determine the domain of each function described. $$ h(z)=-\sqrt[6]{5 z+2} $$

Short Answer

Expert verified
The domain is \( z \in [-\frac{2}{5}, \, \infty) \).

Step by step solution

01

Understanding the Function

The given function is \( h(z) = -\sqrt[6]{5z + 2} \). This is a sixth root function, which means we'll need to consider the domain of the radicand (the expression inside the radical), which is \( 5z + 2 \).
02

Identifying the Domain

For an even-indexed root (like 6 in this case), the radicand must be non-negative (i.e., \( 5z + 2 \geq 0 \)). Solve this inequality for \( z \).
03

Solving the Inequality

Rearrange the inequality: \( 5z + 2 \geq 0 \). Subtract 2 from both sides to get \( 5z \geq -2 \). Divide both sides by 5: \( z \geq -\frac{2}{5} \).
04

Expressing the Domain

The domain is all real numbers \( z \) such that \( z \) is greater than or equal to \( -\frac{2}{5} \). This can be written in interval notation: \( z \in [-\frac{2}{5}, \, \infty) \).

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Key Concepts

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

radicals
Radicals, often referred to as roots, represent the opposite operation to exponentiation. The expression \( \sqrt[n]{a} \) reads as the \

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