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Chapter 12: Problem 8

Classify each of the following statements as either true or false. The domain of the function given by \(f(x)=\ln (x+2)\) is \((-2, \infty)\)

Short Answer

Expert verified
True.

Step by step solution

01

- Understand the Natural Logarithm Domain

Recall that the domain of the natural logarithm function, \(\text{ln}(x)\), consists of all positive real numbers. In other words, \(\text{ln}(x)\) is defined for \(x > 0\).
02

- Analyze the Inside of the Logarithm

In this problem, the function is given by \(f(x) = \text{ln}(x + 2)\). To ensure the input to the logarithm is positive, \(x + 2 > 0\) must hold true.
03

- Solve the Inequality

Solve the inequality \(x + 2 > 0\) to find the domain. Subtract 2 from both sides to get \[ x > -2 \].
04

- Express the Domain in Interval Notation

The inequality \ (x > -2) \ can be written in interval notation as \((-2, \infty)\).
05

- Classify the Statement

The provided statement says that the domain of \(f(x)=\text{ln}(x+2)\) is \((-2, \infty)\). Since our result matches the statement, it is true.

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Key Concepts

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

Understanding Natural Logarithms
The natural logarithm, denoted as \(\text{ln}(x)\), is a special logarithm with the base 'e' (around 2.71828). It is only defined for positive numbers. So, \(\text{ln}(x)\) exists only when \(x > 0\).
For our function \(f(x) = \text{ln}(x+2)\), we need the expression inside the logarithm \((x+2)\) to be greater than zero to make sure \(\text{ln}\) is defined. This gives the inequality \(x+2 > 0\).
Solving Inequalities
Inequalities allow us to find a range of possible solutions. To solve \(x + 2 > 0\), we simply isolate \(x\).
Subtract 2 from both sides to get: \(\text{x + 2 - 2 > 0 - 2}\), simplifying to: \(x > -2\).
This tells us that \(\text{ln}(x+2)\) is defined for any value of \(x\) greater than -2.
Using Interval Notation
Interval notation is a shorthand way of writing expressions for sets defined by inequalities.
For the inequality \(x > -2\), the interval notation is \((-2, \infty)\).
It reads as 'all numbers greater than -2 and less than infinity.' The round bracket means -2 is not included in the set. On the other hand, when a number is included, we use a square bracket.
Connecting Algebra to Domain
To find the domain of a function involving a natural logarithm, follow these steps:
  • Identify the expression inside the logarithm.
  • Set up an inequality where this expression is greater than zero.
  • Solve the inequality to find valid \(x\) values.
  • Express this solution in interval notation.

For our function \(f(x) = \text{ln}(x+2)\), solving \(x+2>0\) gives us \(x > -2\). The interval notation for this solution is \((-2, \infty)\). Thus, the domain of \(f(x)\) is indeed \((-2, \infty)\), making the statement true.

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