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Chapter 2: Problem 14

Express the domain of the given function using interval notation. $$ f(x)=\ln \left(x^{2}-4\right) $$

Short Answer

Expert verified
The domain of the function is \((-\infty, -2) \cup (2, \infty)\).

Step by step solution

01

Understand the Function

The function given is a logarithmic function: \[ f(x) = \ln(x^2 - 4) \] Logarithmic functions are defined only for positive arguments, so we need \( x^2 - 4 > 0 \). This means \( x^2 > 4 \).
02

Solve the Inequality

To solve \( x^2 > 4 \), consider 1. \( x > 2 \) 2. \( x < -2 \) These are the values of \( x \) that make the inequality true.
03

Express the Domain in Interval Notation

The solutions from the inequality \( x > 2 \) and \( x < -2 \) tell us the intervals where the function is defined. In interval notation, the domain is: \[ (-\infty, -2) \cup (2, \infty) \] This combines all \( x \) values where the function is valid.

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Key Concepts

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

Interval Notation
Interval notation is a mathematical notation used to describe a set of numbers between given bounds. It provides a clear and concise way to express the range of numbers a function can take on. For example, when we express the domain of the function \( f(x) = \ln(x^2 - 4) \), we want to show which \( x \) values make \( x^2 - 4 \) positive.
  • The symbol \((-\infty, -2)\) represents all numbers less than \(-2\), extending to negative infinity.
  • \((2, \infty)\) shows numbers greater than \(2\), extending to positive infinity.
The union symbol \(\cup\) is used between intervals to signify that either interval can be a valid set of numbers for the domain of \( f(x) \). Interval notation is preferred in mathematics because it is simple and eliminates the need for lengthy descriptions.
Inequality Solutions
Inequality solutions help us find the range of values that satisfy certain conditions. For the function \( f(x) = \ln(x^2 - 4) \), the inequality is \( x^2 - 4 > 0 \). To solve this, we need to find values of \( x \) that make this statement true.
  • We first rearrange to get \( x^2 > 4 \).
  • This can be split into two separate inequalities: \( x > 2 \) and \( x < -2 \).
Why two inequalities? Because squared numbers are always positive, so \( x \) needs to be either greater than \(2\) or less than \(-2\). The solvable parts of these inequalities help determine the function's domain.
Function Domain
The domain of a function is the set of "input" values that the function can accept without causing any undefined behavior. For logarithmic functions, like \( f(x) = \ln(x^2 - 4) \), it's crucial to remember that the argument inside the log must be positive. Therefore, the domain depends on where this condition holds true.
  • The function should not receive any input that makes the argument of the logarithm less than or equal to zero.
  • Based on \( x^2 - 4 > 0 \), solutions are achieved for the specific \( x \) values such as \( x > 2 \) and \( x < -2 \), confirming the function's domain in those ranges.
This ensures the logarithmic functions operate without error and are well-defined for given values.
Logarithmic Function
Logarithmic functions, such as \( f(x) = \ln(x^2 - 4) \), are used in mathematics to describe relationships involving multiplicative growth. An essential property of logarithmic functions is they are only defined for positive real numbers.
  • If you try to take the logarithm of zero or a negative number, the result is undefined in the realm of real numbers.
  • This is why the argument \( x^2 - 4 \) must be positive, as finding the logarithm of a non-positive number would make the function invalid.
Understanding this principle is key for identifying the domain of logarithmic functions and ensures they're properly graphed and analyzed.

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