Problem 16 - College Algebra with Corequisite Support

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: \(f(x) \to \infty\) as \(x \to -\infty\), \(f(x) \to -\infty\) as \(x \to 2^-\).

Step 1: Determine the Domain

The domain of the function is the set of all values of \(x\) for which the function is defined. The logarithm \(\ln(y)\) is defined for \(y > 0\). Here, \(y = (2 - x)\), thus we have \(2 - x > 0\). By solving this inequality, we get \(x < 2\). So the domain of the function is \((-\infty, 2)\).

Step 2: Identify the Vertical Asymptote

Vertical asymptotes occur at values of \(x\) that make the inside of the logarithm zero because the logarithm becomes undefined. Set \(2-x = 0\) to find the x-value. Solving \(x = 2\) gives the vertical asymptote at \(x = 2\).

Step 3: Analyze the End Behavior

End behavior refers to the behavior of \(f(x)\) as \(x\) approaches the endpoints of its domain. As \(x \to -\infty\), \(f(x) = \ln(2-x)\to \ln(\infty) = \infty\), meaning \(f(x)\) increases without bound. As \(x\) approaches 2 from the left (\(x \to 2^-\)), \(f(x) = \ln(2-x)\) approaches \(\ln(0) = -\infty\), meaning \(f(x)\) decreases without bound.

Key Concepts

Vertical Asymptotes

Logarithmic functions like \(f(x) = \ln(2-x)\) have interesting features that many students find challenging to understand. One such feature is the vertical asymptote. A vertical asymptote is a vertical line, \(x = a\), where a function's value grows very large or very small as it approaches \(a\). This is because at the vertical asymptote, the expression inside the logarithm becomes zero or negative, which is undefined in logarithms.
\[ \text{For } f(x) = \ln(2-x), \text{ you set the inside } (2-x) = 0.\]
Solving this gives \(x = 2\), the location of our vertical asymptote.

• At \(x = 2\): The graphical representation shows that the function approaches negative infinity, meaning the graph gets closer to the line \(x = 2\) without touching it.
• Understanding vertical asymptotes helps you predict drastic changes in function values as you near specific x-values.

End Behavior of Functions

The end behavior of a function helps you understand how the function behaves as the input values head towards the extreme, either towards positive or negative infinity. In the logarithmic function \(f(x) = \ln(2-x)\), the end behavior is apparent as \(x\) approaches the endpoints of its domain, \((-\infty, 2)\).

As \(x\) approaches negative infinity:

• Substitute very small values into \(2-x\) and note that the function approaches \(\ln(\infty) = \infty\).
• This suggests that \(f(x)\) increases without bound as \(x\) decreases.

As \(x\) approaches 2 from the left (written as \(x \to 2^-\)):

• The expression \(2-x\) turns into a small positive number, causing \(\ln(2-x)\) to decrease rapidly.
• This is why \(f(x)\) dashes towards \(-\infty\), indicating a steep descent on the graph.

Grasping this end behavior is crucial for analyzing, sketching, and predicting the logarithmic function's performance across its domain.

Logarithmic Inequalities

Logarithmic inequalities play an essential role in determining the domain of logarithmic functions and understanding features like vertical asymptotes. When we encounter \(f(x) = \ln(2-x)\), we're tasked with ensuring that the argument \((2-x)\) inside the logarithm always stays positive since \(\ln(y)\) is defined only if \(y > 0\).

To find the domain:

• Set \(2-x > 0\).
• Solve the inequality to obtain \(x < 2\), meaning the domain is \((-\infty, 2)\).

Practicing logarithmic inequalities will help you:

• Acquire the skill to solve inequalities systematically to determine valid input values that keep the function defined.
• Prevent common mistakes involving assumptions that logarithms can take negative values.
• Improve your overall understanding of both basic and complex logarithmic functions.

Logarithmic inequalities are foundational for analyzing and understanding the limits and behavior of their related functions.