Problem 17 - College Algebra with Corequisite Support

Question

For the following exercises, state the domain, vertical asymptote, and end behavior of the function. \(f(x)=\log \left(x-\frac{3}{7}\right)\)

Step-by-Step Solution

Answer

Domain is \( x > \frac{3}{7} \), vertical asymptote at \( x = \frac{3}{7} \), and end behavior is \( f(x) \to \infty \) as \( x \to \infty \), \( f(x) \to -\infty \) as \( x \to \frac{3}{7}^+ \).

Step 1: Determine the Domain of the Function

The domain of the function is where the expression inside the logarithm is positive. Therefore, solve the inequality: \[ x - \frac{3}{7} > 0 \]Add \(\frac{3}{7}\) to both sides:\[ x > \frac{3}{7} \]Thus, the domain of the function is \( x > \frac{3}{7} \).

Step 2: Identify the Vertical Asymptote

The vertical asymptote occurs where the expression inside the logarithm is equal to zero:\[ x - \frac{3}{7} = 0 \]Solving gives:\[ x = \frac{3}{7} \]Therefore, the vertical asymptote is at \( x = \frac{3}{7} \).

Step 3: Examine the End Behavior of the Function

As \( x \to \infty \), \( f(x) \to \infty \) because the logarithm continues to increase as \( x \) grows larger. Additionally, as \( x \to \frac{3}{7}^+ \), \( f(x) \to -\infty \) because the logarithm approaches negative infinity as the input approaches the vertical asymptote from the right.

Key Concepts

Understanding The Function Domain of Logarithmic Functions

The domain of a function is a crucial concept that tells us all the possible input values (or "x" values) that will yield a real number as an output for the function. For logarithmic functions, like the one we are discussing, the domain is defined by the set of values for which the argument of the logarithm is greater than zero. In the case of the function \( f(x) = \log \left( x - \frac{3}{7} \right) \), the expression inside the logarithm is \( x - \frac{3}{7} \). To find the domain, set this greater than zero:

• \( x - \frac{3}{7} > 0 \)
• Add \( \frac{3}{7} \) to both sides to get \( x > \frac{3}{7} \)

This means the domain is all real numbers greater than \( \frac{3}{7} \). Therefore, \( x \) has to be greater than \( \frac{3}{7} \) for the logarithmic function to be defined. Understanding this ensures we know which values make our function work without running into mathematical errors.

Exploring Vertical Asymptotes in Logarithmic Functions

Vertical asymptotes are lines that the graph of a function approaches but never actually touches or crosses. In a logarithmic function, a vertical asymptote occurs where the logarithm is undefined. This occurs when the argument of the logarithm is zero. For our function \( f(x) = \log \left( x - \frac{3}{7} \right) \), finding the vertical asymptote involves setting:

• \( x - \frac{3}{7} = 0 \)
• Solve to find \( x = \frac{3}{7} \)

This tells us there is a vertical asymptote at \( x = \frac{3}{7} \).

What This Means for the Graph

The vertical asymptote at \( x = \frac{3}{7} \) indicates that as we get closer to this value from the right, the function \( f(x) \) heads towards negative infinity. The function does not exist at \( x = \frac{3}{7} \) and dramatically drops as it approaches this value, which is why it cannot be crossed or even touched by the graph.

End Behavior of Logarithmic Functions

End behavior in the context of a function refers to the direction the function values move as the input \( x \) approaches extreme values, primarily infinity or near an asymptote. For the logarithmic function \( f(x) = \log \left(x - \frac{3}{7}\right) \), understanding the end behavior gives us an idea of how the graph behaves for very large \( x \) and as \( x \) approaches our vertical asymptote from the right.

As \( x \to \infty \)

As \( x \) increases towards infinity, the function \( f(x) \) also rises towards infinity. The logarithm continues to increase as there is no bound to how large \( x \) can grow.

Approaching the Asymptote

As \( x \to \frac{3}{7}^+ \), where \( \frac{3}{7}^+ \) implies approaching from the right of \( \frac{3}{7} \), the function heads toward negative infinity. The closer \( x \) is to \( \frac{3}{7} \) from the right, the more negative \( f(x) \) becomes, illustrating the function's sharp decline as we near the vertical asymptote.