Problem 17 - College Algebra

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: \(x > \frac{3}{7}\); Vertical Asymptote: \(x = \frac{3}{7}\); End Behavior: \(f(x) \to \infty\) as \(x \to \infty\).

Step 1: Finding the Domain

The domain of a logarithmic function is determined by the expression inside the logarithm. For the function \(f(x) = \log\left(x - \frac{3}{7}\right)\), the expression \(x - \frac{3}{7}\) must be greater than zero. This gives us the inequality: \(x - \frac{3}{7} > 0\). Solving for \(x\), we add \(\frac{3}{7}\) to both sides, resulting in \(x > \frac{3}{7}\). Thus, the domain of the function is \(x > \frac{3}{7}\).

Step 2: Identifying the Vertical Asymptote

A vertical asymptote in a logarithmic function occurs at the value that makes the expression inside the logarithm equal to zero. For \(f(x) = \log\left(x - \frac{3}{7}\right)\), setting \(x - \frac{3}{7} = 0\) and solving for \(x\) gives \(x = \frac{3}{7}\). Therefore, the vertical asymptote is located at \(x = \frac{3}{7}\).

Step 3: Analyzing the End Behavior

The end behavior of the function describes what happens to \(f(x)\) as \(x\) approaches infinity. For \(f(x) = \log\left(x - \frac{3}{7}\right)\), as \(x\) approaches infinity, the expression \(x - \frac{3}{7}\) also approaches infinity. Since the logarithm of a large positive number is also large, \(f(x)\) approaches infinity. Formally, \(\lim_{{x \to \infty}} f(x) = \infty\).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (usually represented by \(x\)) that will produce a valid output when plugged into the function. For logarithmic functions, the expression inside the logarithm must always be greater than zero because logarithms of non-positive numbers are undefined in the real number system.

To find the domain of the function \(f(x) = \log(x - \frac{3}{7})\), we are interested in where the expression \(x - \frac{3}{7}\) is greater than zero. This leads us to the inequality:

• \(x - \frac{3}{7} > 0\)

We solve this inequality by adding \(\frac{3}{7}\) to both sides, resulting in:

• \(x > \frac{3}{7}\)

Therefore, the domain is all real numbers greater than \(\frac{3}{7}\), or expressed in interval notation, \((\frac{3}{7}, \infty)\). This tells us that any \(x\) value larger than \(\frac{3}{7}\) can be used in our original function without causing any mathematical issues.

Vertical Asymptote

A vertical asymptote is a characteristic of specific types of functions where the function approaches infinity as it nears a particular \(x\)-value but never actually reaches it. In the context of logarithmic functions, vertical asymptotes occur where the argument inside the log function equals zero because the log of zero is undefined.

For the function \(f(x) = \log(x - \frac{3}{7})\), we determine the vertical asymptote by setting the expression inside the logarithm equal to zero:

• \(x - \frac{3}{7} = 0\)

By solving this equation, we find:

• \(x = \frac{3}{7}\)

This means there is a vertical asymptote at \(x = \frac{3}{7}\). The function gets closer and closer to this line but never intersects or crosses it. This is crucial for understanding the behavior and constraints of the function.

End Behavior of Functions

The end behavior of a function describes how the function behaves as the input \(x\) approaches very large positive or negative values (typically towards infinity or negative infinity). For different types of functions, this behavior can vary significantly.

In logarithmic functions, such as \(f(x) = \log(x - \frac{3}{7})\), we are interested in the direction that the function heads as \(x\) becomes very large. Specifically, as \(x\) approaches infinity, the expression inside the log function \(x - \frac{3}{7}\) also grows without bound. The properties of logarithms tell us that:

• As the argument of the logarithm increases indefinitely, the output (or \(f(x)\)) also increases indefinitely.

Therefore, as \(x\) goes to infinity, \(f(x)\) also approaches infinity. Formally, this is expressed as:

• \(\lim_{{x \to \infty}} f(x) = \infty\)

Recognizing this behavior helps understand how the function behaves at its extremes, which is important for graphing and analyzing the function completely.