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Chapter 10: Problem 102

Determine the domain of each function described. $$ d(x)=-\sqrt[4]{7 x-5} $$

Short Answer

Expert verified
The domain is \( x \geq \frac{5}{7} \).

Step by step solution

01

- Understand the function

The function given is \( d(x) = -\sqrt[4]{7x - 5} \). The expression inside the fourth root must be non-negative since the fourth root of a negative number is not a real number.
02

- Set up the inequality

To find the domain, set the expression \(7x - 5\) greater than or equal to zero: \(7x - 5 \geq 0 \).
03

- Solve the inequality

Solve the inequality \(7x - 5 \geq 0\): \(7x \geq 5\). Divide both sides by 7: \(x \geq \frac{5}{7}\).
04

- Write the domain

The domain of the function \(d(x)\) is all real numbers \(x\) such that \( x \geq \frac{5}{7} \).

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Key Concepts

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

inequalities
Inequalities are mathematical expressions involving the symbols >, <, 鈮�, and 鈮�. These symbols compare two values or expressions.
For example, in the inequality 7x - 5 鈮� 0, we are stating that 7x - 5 must be greater than or equal to 0.
By changing the expression to an inequality, we identify the range of values that satisfy the condition within the function. Inequalities are crucial in functions where certain values might not make sense or be permissible, such as negative values inside a square root or fourth root.
fourth root
The fourth root of a number is the value that, when multiplied by itself four times, gives the original number. For example, the fourth root of 16 is 2 because 2 脳 2 脳 2 脳 2 = 16.
In the given function, we have a fourth root: \( d(x) = -\sqrt[4]{7x - 5} \).
It's important to note that the expression inside the fourth root, 7x - 5, needs to be non-negative (greater than or equal to zero).
This requirement arises because the fourth root of a negative number is not defined within the realm of real numbers.
solving inequalities
When solving inequalities, the goal is to isolate the variable on one side of the inequality to find the solution set.
For our function, we set up the inequality 7x - 5 鈮� 0 to ensure that the fourth root expression is non-negative.
Here's how to solve it step-by-step:
  • Start with the inequality: 7x - 5 鈮� 0.
  • Add 5 to both sides: 7x 鈮� 5.
  • Divide both sides by 7: \( x \geq \frac{5}{7} \).

Thus, the solution is x must be greater than or equal to 5/7. This solution tells us the domain of the function since these are the x-values that keep the expression inside the fourth root non-negative.
real numbers
Real numbers include all the numbers on the number line. This includes both positive and negative numbers, whole numbers, fractions, and irrational numbers.
In mathematics, real numbers are often denoted by the symbol 鈩�.
In the context of our function, the domain consists of all real numbers x such that x 鈮� 5/7. This means all real numbers starting from 5/7 onwards are valid inputs for our function.
Understanding the concept of real numbers is crucial because certain mathematical operations require the results to be real numbers to make sense and be usable within specific contexts.

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Most popular questions from this chapter

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