Problem 205 - Precalculus

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

The domain is \( (\frac{3}{7}, \infty) \), the vertical asymptote is \( x = \frac{3}{7} \), and as \( x \to \infty \), \( f(x) \to \infty \).

Step 1: Determine the Domain

The domain of a function is the set of all possible input values for which the function is defined. For the logarithm function \( f(x) = \log\left(x - \frac{3}{7}\right) \), the argument of the logarithm (i.e., \( x - \frac{3}{7} \)) must be greater than zero. Therefore, we solve the inequality: \[ x - \frac{3}{7} > 0 \] Solving this gives: \[ x > \frac{3}{7} \]. Thus, the domain of the function is \( (\frac{3}{7}, \infty) \).

Step 2: Identify the Vertical Asymptote

Vertical asymptotes occur where the function approaches infinity or negative infinity as it nears a specific point, typically due to division by zero or a logarithmic undefined value. For \( f(x) = \log\left(x - \frac{3}{7}\right) \), the logarithmic function is undefined at the point \( x = \frac{3}{7} \). Thus, \( x = \frac{3}{7} \) is the vertical asymptote.

Step 3: Analyze End Behavior

End behavior describes how the function behaves as \( x \to \infty \) or \( x \to -\infty \). Since the domain of \( f(x) \) is \( (\frac{3}{7}, \infty) \), we only consider \( x \to \infty \). As \( x \to \infty \), the expression \( x - \frac{3}{7} \to \infty \), and thus \( \log(x - \frac{3}{7}) \to \infty \). Therefore, as \( x \to \infty \), \( f(x) \to \infty \).

Key Concepts

Domain of Logarithmic Function

When dealing with logarithmic functions, the domain is crucial because it defines the set of allowable input values that keep the function valid. For example, consider the function \( f(x) = \log\left(x - \frac{3}{7}\right) \). The key point to remember is that for a logarithm \( \log(a) \), the argument \( a \) must be greater than zero.
In our case, the argument is \( x - \frac{3}{7} \). We need this to be greater than zero, leading to the inequality \( x - \frac{3}{7} > 0 \). Solving this inequality is straightforward:

• Add \( \frac{3}{7} \) to both sides to isolate \( x \).
• The solution is \( x > \frac{3}{7} \).

So, the domain of the function is all values greater than \( \frac{3}{7} \), or \((\frac{3}{7}, \infty)\) using interval notation.

Vertical Asymptote

Vertical asymptotes are lines that the graph of a function approaches but never touches or crosses. They arise when the function becomes undefined at certain points, causing the function's value to shoot to infinity, either positively or negatively.
For the function \( f(x) = \log\left(x - \frac{3}{7}\right) \), we establish a vertical asymptote by finding where the function is undefined. Since logarithms are undefined at zero or negative values, set the argument \( x - \frac{3}{7} = 0 \) to find the undefined point:

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

Thus, \( x = \frac{3}{7} \) is a vertical asymptote. As \( x \) approaches \( \frac{3}{7} \) from the right, the function's values trend towards negative infinity.

End Behavior of Functions

End behavior refers to how a function behaves as the input \( x \) becomes very large (\( x \to \infty \)) or very small (\( x \to -\infty \)). It's like predicting the trend of the graph at its extreme ends.
For \( f(x) = \log\left(x - \frac{3}{7}\right) \), we concern ourselves with the behavior as \( x \to \infty \), since the function's domain is only valid for \( x > \frac{3}{7} \).

• As \( x \to \infty \), the expression \( x - \frac{3}{7} \) tends to infinity.
• And naturally, \( \log(x - \frac{3}{7}) \to \infty \) as well.

Consequently, as \( x \to \infty \), \( f(x) \) goes to infinity too. This tells us that the graph of the function keeps rising as it moves further to the right on the \( x \)-axis.