Problem 199 - Precalculus

Question

For the following exercises, state the domain and the vertical asymptote of the function. $$f(x)=\log _{b}(x-5)$$

Step-by-Step Solution

Answer

The domain is \((5, \infty)\) and the vertical asymptote is at \(x=5\).

Step 1: Understand the Function

The given function is \( f(x) = \log_b(x-5) \). This is a logarithmic function, which generally has the form \( \log_b(u) \), where \( u \) is the argument of the logarithm.

Step 2: Determine the Domain

The domain of a logarithmic function like \( \log_b(u) \) is determined by the condition that the argument \( u \) must be positive. So for \( \log_b(x-5) \), we require \( x - 5 > 0 \). Solve this inequality to find the domain: \( x > 5 \). Thus, the domain is \( (5, \infty) \).

Step 3: Identify the Vertical Asymptote

Vertical asymptotes occur in logarithmic functions where the argument approaches zero, leading the function to approach negative infinity. For \( \log_b(x-5) \), this happens as \( x \) approaches 5 from the right. Therefore, the vertical asymptote is at \( x = 5 \).

Key Concepts

Domain of a Function

The domain of a function represents the set of all possible input values (or x-values) for which the function is defined. In the context of logarithmic functions, understanding the domain is crucial, as the function requires certain conditions to be met for it to exist.

For a logarithmic function like \( f(x) = \log_b(x-5) \), the input to the logarithmic function, which is expressed as \( u = x - 5 \), must always be a positive number. That's because the logarithm of a non-positive number is undefined in the realm of real numbers. To find the domain, solve the inequality \( x - 5 > 0 \).

Here's how you find the domain:

• Solve the inequality: \( x - 5 > 0 \).
• Adding 5 to both sides gives \( x > 5 \).
• Therefore, the domain in interval notation is \( (5, \infty) \).

So, for the function to "work," you can plug in any x-value greater than 5.

Vertical Asymptote

A vertical asymptote in a function is a line where the function heads towards infinity or negative infinity. It represents a value that x can approach but never actually reach, often causing the function itself to become undefined at that point.

In the case of the logarithmic function \( f(x) = \log_b(x-5) \), a vertical asymptote appears when the argument of the logarithmic function approaches zero, i.e., \( x - 5 = 0 \).

Here's a closer look at finding a vertical asymptote:

• Identify when the logarithm's argument equals zero: \( x - 5 = 0 \).
• Solving this equation, you find \( x = 5 \).
• The vertical asymptote is therefore at \( x = 5 \).

The graph of this function will approach the line \( x = 5 \) but will never touch or cross it.

Inequalities in Mathematics

Inequalities in mathematics are statements about the relative size or order of two objects. When dealing with functions, inequalities are often used to determine domains, ranges, and other constraints that define the behavior of the function.

In solving the domain for the function \( f(x) = \log_b(x-5) \), inequalities played a crucial role. The inequality \( x - 5 > 0 \) needed to be solved to find the domain of the function.
Here's a brief overview of tackling inequalities:

• Write the inequality based on the conditions needed for the function.
• Solve the inequality using algebraic manipulation, such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same number.
• Remember that multiplying or dividing by a negative number flips the inequality sign.

By effectively using inequalities, you can determine critical attributes of a function, such as its domain and intervals where the function is increasing or decreasing.