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Magazine Chapter 4: Problem 15
Find the domain and range of each function: $$ f(x)=\ln \left(x^{2}-1\right) $$
Short Answer
Expert verified
The domain is \((-\infty, -1) \cup (1, \infty)\), and the range is \((-\infty, \infty)\).
Step by step solution
01
Determine the Domain Conditions
The function is a natural logarithm, which means it is only defined for positive arguments. Therefore, we need to solve the inequality \(x^2 - 1 > 0\) to find the domain.
02
Solve the Inequality
Solve the inequality \(x^2 - 1 > 0\).First, factor the inequality:\(x^2 - 1 = (x-1)(x+1)\).This implies that \((x - 1)(x + 1) > 0\).
03
Test Intervals
To find where the product is positive, consider the intervals defined by the roots: \((-\infty, -1)\), \((-1, 1)\), and \((1, \infty)\).- For \(x \in (-\infty, -1)\), both factors \((x-1)\) and \((x+1)\) are negative, so their product is positive.- For \(x \in (-1, 1)\), \((x-1)\) is negative and \((x+1)\) is positive, so their product is negative.- For \(x \in (1, \infty)\), both factors are positive, so their product is positive.
04
Combine Valid Intervals for Domain
The inequality \(x^2 - 1 > 0\) is satisfied in the intervals \((-\infty, -1)\) and \((1, \infty)\). Therefore, the domain of \(f(x) = \ln(x^2 - 1)\) is the union of these intervals: \((-\infty, -1) \cup (1, \infty)\).
05
Determine the Range of the Function
Since the natural logarithm can take any real number as a result, and the inputs \(x^2 - 1\) can vary from just above zero to infinity for \(x \in (-\infty, -1) \cup (1, \infty)\), the range of \(f(x)\) is \((-\infty, \infty)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Inequality Solving
Inequality solving is a fundamental process in mathematics, especially when we deal with domains of functions. To find the domain of a given function, we often need to solve an inequality. This involves determining which values of the variable satisfy the condition, making the expression or function meaningful.
Here's a typical approach to solving inequalities:
Here's a typical approach to solving inequalities:
- First, isolate the expression of interest (e.g., factor it if possible).
- Identify the critical points where the expression changes sign, such as the roots for quadratic expressions.
- Split the number line into intervals based on these critical points.
- Test each interval to see if it satisfies the inequality.
- Finally, select the intervals that meet the inequality conditions.
Diving into Natural Logarithms
The natural logarithm, denoted as \(\ln(x)\), is a logarithm to the base of the mathematical constant \(e\). This constant \(e\) is approximately equal to 2.71828. A natural logarithm is distinct from other logarithms because it is specifically used in higher mathematics due to its natural properties, particularly in calculus and analysis.
Key properties of the natural logarithm include:
Key properties of the natural logarithm include:
- Its domain: the function \(\ln(x)\) is only defined for positive \(x\). This means you can only take the natural log of positive numbers.
- Its range: \(\ln(x)\) can produce all real numbers, spanning from \(-\infty\) to \(\infty\). This allows the function to cover a wide range when solving equations.
- The derivative and integral, essential in calculus for finding slopes and areas under curves.
Insights into Function Analysis
Function analysis involves examining various aspects of a function to understand its behavior. This includes looking at the domain and range, intervals where the function increases or decreases, and points of interest such as maxima and minima.
When analyzing a function like \(f(x) = \ln(x^2 - 1)\), it's crucial to:
When analyzing a function like \(f(x) = \ln(x^2 - 1)\), it's crucial to:
- Identify the domain: This limits the values you can input into the function. In our case, it's \((-\infty, -1) \cup (1, \infty)\).
- Determine the range: Knowing the outputs the function can produce helps understand its influence. For natural logs, this is typically \(( -\infty, \infty)\).
- Examine the shape and direction of the function: This includes looking at graphs and calculating derivatives, though beyond the basic scope discussed here.
