Problem 11 - College Algebra

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

Domain: \((5, \infty)\); Vertical asymptote: \(x=5\).

Step 1: Identify the Domain Condition

The function in question is a logarithmic function of the form \( f(x) = \log_b(x-5) \). For the logarithmic function \( \log_b(u) \), the argument \( u \) must be positive. Therefore, set \( x - 5 > 0 \).

Step 2: Solve the Inequality

Solve the inequality \( x - 5 > 0 \) to find the domain of the function. Add 5 to both sides to get: \( x > 5 \). Therefore, the domain of the function is \( x > 5 \), or expressed in interval notation: \((5, \infty)\).

Step 3: Determine the Vertical Asymptote

A vertical asymptote occurs when the input of the logarithmic function approaches a value from the left within its domain. This happens when the argument of the log goes to zero. Set \( x - 5 = 0 \) to find the vertical asymptote. Solving gives \( x = 5 \). Thus, the vertical asymptote is the line \( x = 5 \).

Key Concepts

Domain of a Function

In mathematics, the domain of a function is the set of all possible input values (usually represented by the variable \( x \)) for which the function is defined. For logarithmic functions like \( f(x) = \log_b(x - 5) \), the domain is determined by the requirement that the argument of the logarithm must be positive. This makes sense because you can't take the logarithm of a non-positive number, as it doesn't yield a real number.

To determine the domain for \( f(x) = \log_b(x - 5) \), we set up the condition \( x - 5 > 0 \). Solving this inequality involves simply adding 5 to both sides, giving us \( x > 5 \). This means the domain of the function in interval notation is \((5, \infty)\).

• Logarithmic functions have domains restricted to positive arguments.
• For \( \log_b(x - 5) \), the domain is \( x > 5 \).
• This ensures that the logarithm function is properly defined for all \( x \) in the domain.

Vertical Asymptotes

A vertical asymptote of a function is a vertical line \( x = a \) where the function approaches infinity as \( x \) gets closer to \( a \) from either side. In the context of the function \( f(x) = \log_b(x - 5) \), a vertical asymptote occurs at the value of \( x \) that makes \( x - 5 = 0 \).

To find this, we solve \( x - 5 = 0 \), which gives us \( x = 5 \). This tells us that \( y \to \infty \) as \( x \) approaches 5 from the right. Hence, there is a vertical asymptote at \( x = 5 \).

• Vertical asymptotes occur when the function approaches infinity sharply.
• For \( \log_b(x - 5) \), the vertical asymptote is at \( x = 5 \).
• They are critical in understanding the behavior of logarithmic functions near domain limits.

Inequalities in Algebra

Understanding and solving inequalities is a vital part of algebra, particularly when dealing with domains of functions. In the case of the domain of a logarithmic function \( f(x) = \log_b(x - 5) \), we use inequalities to determine the values of \( x \) where the function is defined.

An inequality like \( x - 5 > 0 \) expresses the condition that must be satisfied for the logarithmic expression to be evaluated. By adding 5 to both sides of the inequality, we determine that \( x > 5 \). This inequality reveals the domain of the function in our example.

• Inequalities help determine valid input ranges or domains for functions.
• Solving \( x - 5 > 0 \) gives \( x > 5 \), defining the domain for the logarithmic function.
• Understanding how to manipulate and solve inequalities is crucial for tackling domain-related problems.