Problem 16 - College Algebra

Question

For the following exercises, state the domain, vertical asymptote, and end behavior of the function. $$f(x)=\ln (2-x)$$

Step-by-Step Solution

Answer

Domain: \((-\infty, 2)\); Vertical Asymptote: \(x = 2\); End Behavior: as \(x \to 2^-\), \(f(x) \to -\infty\); as \(x \to -\infty\), \(f(x) \to \infty\).

Step 1: Identify the Domain of the Function

To find the domain of the function \(f(x) = \ln(2-x)\), we need to ensure the argument of the natural logarithm is positive, because the logarithm is only defined for positive numbers. So, the inequality \(2-x > 0\) must be satisfied. Solving \(2-x > 0\), we get \(x < 2\). Therefore, the domain of the function is all real numbers \(x\) such that \(x < 2\). In interval notation, this is \((-\infty, 2)\).

Step 2: Determine the Vertical Asymptote

Vertical asymptotes occur where the function is undefined or the function approaches \(\infty\) or \(-\infty\). For \(f(x) = \ln(2-x)\), the function becomes undefined when its argument is zero, i.e., at \(x = 2\). Therefore, \(x = 2\) is where the vertical asymptote occurs.

Step 3: Analyze the End Behavior of the Function

To understand the end behavior of \(f(x) = \ln(2-x)\), we consider values of \(x\) approaching the domain boundaries. As \(x\) approaches \(2\) from the left (\(x \to 2^-\)), \(2-x\) approaches \(0^+\), making \(\ln(2-x)\) approach \(-\infty\). For \(x\) approaching \(-\infty\), \(f(x)\) behaves like \(\ln(\infty)\), which goes to \(\infty\). Thus, as \(x \to 2^-\), \(f(x) \to -\infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).

Key Concepts

Natural Logarithm

The natural logarithm is a mathematical function denoted as \( \ln(x) \). It is the inverse of the exponential function with base \( e \), which is approximately equal to 2.718. The key idea behind the natural logarithm is to determine the power to which \( e \) must be raised to obtain a given number.

• For example, \( \ln(e) = 1 \) because \( e^1 = e \).
• Similarly, \( \ln(1) = 0 \) because \( e^0 = 1 \).

Keep in mind that the natural logarithm is only defined for positive numbers. This restriction is crucial when determining the domain of functions involving \( \ln(x) \). For \( f(x) = \ln(2-x) \), the expression \( 2-x \) must be greater than zero for \( f(x) \) to be defined, leading to \( x < 2 \) as the domain.

Vertical Asymptote

A vertical asymptote is a vertical line \( x = a \) where a function increases or decreases without bound as it approaches \( a \). It indicates a value that the function cannot take but gets infinitely close to.In the case of \( f(x) = \ln(2-x) \), the vertical asymptote occurs at \( x = 2 \) because the function becomes undefined there. As \( x \) gets closer and closer to 2 from the left, the expression inside the logarithm \( 2-x \) gets very close to zero. Since \( \ln(0) \) is undefined and the logarithm approaches negative infinity, \( x = 2 \) is thus a vertical asymptote for the function. Understanding where vertical asymptotes occur helps in graphing functions and predicting behavior as inputs approach certain critical values.

End Behavior

End behavior describes how a function behaves as the input \( x \) approaches extremely large or small values. It gives us insight into what happens as \( x \) moves towards the boundaries of its domain or continues to grow larger or smaller.For \( f(x) = \ln(2-x) \), the end behavior analysis involves observing \( f(x) \) as \( x \) approaches the edges of its domain.

• As \( x \to 2^- \) (approaching 2 from the left), \( 2-x \) goes to zero from the positive side, causing \( f(x) \) to go to \(-\infty\).
• As \( x \to -\infty \), \( 2-x \) becomes a large positive number, making \( \ln(2-x) \) go to \( \infty \).

This exploration of end behavior helps us understand the long-term trends of the function and its overall graph.