Problem 11 - College Algebra with Corequisite Support

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: \(x > 5\); Vertical asymptote: \(x=5\).

Step 1: Understanding the Domain

The function given is a logarithmic function, \(f(x) = \log_b(x-5)\). For a logarithmic function \(\log_b(u)\), the argument \(u\) must be greater than zero. Thus, to find the domain of \(f(x)\), we solve the inequality \(x-5 > 0\).

Step 2: Solving the Inequality for the Domain

Solve the inequality \(x-5 > 0\) to find the domain of the function. Adding 5 to both sides, we get \(x > 5\). Therefore, the domain of the function \(f(x)\) is \(x > 5\).

Step 3: Identifying the Vertical Asymptote

In a logarithmic function \(\log_b(x-a)\), the vertical asymptote occurs at the point where the argument \(x-a=0\). For \(f(x) = \log_b(x-5)\), set \(x-5 = 0\).

Step 4: Solving for the Vertical Asymptote

Solve the equation \(x-5=0\) to find the vertical asymptote. Adding 5 to both sides, we find that the vertical asymptote is at \(x=5\).

Key Concepts

Domain of a Function

When dealing with logarithmic functions, determining the domain is crucial. The domain of a function refers to all the possible input values (x-values) that will yield a valid output value. In a logarithmic function like \(f(x) = \log_b(x-5)\), the argument of the logarithm—\(x-5\) in this case—must be greater than zero. This is because the logarithm of zero or a negative number is not defined in the realm of real numbers. To find the domain, you'll need to solve the inequality \(x-5 > 0\). This involves simple algebraic manipulation. Add 5 to both sides, resulting in \(x > 5\). Thus, the domain of the function \(f(x)\) is all real numbers greater than 5. You can express it in interval notation as \((5, \infty)\). Here’s a quick breakdown:

• Identify the argument of the logarithm. In this function, it's \(x-5\).
• Set up the inequality so that the argument is greater than zero: \(x-5 > 0\).
• Solve the inequality to find the domain: \(x > 5\).

Vertical Asymptote

A vertical asymptote in a function is a value of \(x\) where the function tends to infinity or negative infinity. It represents a value that \(x\) can get very close to, but never actually reach. For logarithmic functions like \(f(x) = \log_b(x-5)\), vertical asymptotes occur where the argument inside the logarithm equals zero.To determine the vertical asymptote, set the argument \(x-5\) equal to zero. Solving the equation \(x-5 = 0\) involves a straightforward step: add 5 to both sides, resulting in \(x = 5\). Therefore, \(x = 5\) is the vertical asymptote of the function.To put it simply:

• Determine the argument of the logarithm function: \(x-5\).
• Set this expression to 0: \(x-5 = 0\).
• Solve for \(x\), finding that the vertical asymptote is at \(x = 5\).

Solving Inequalities

Solving inequalities is a fundamental skill in understanding the behavior of functions, including logarithmic functions. When you solve inequalities, you're looking for a set of values that satisfy the inequality condition, ensuring the function is properly defined.For example, with the function \(f(x) = \log_b(x-5)\), the task was to solve the inequality \(x-5 > 0\). This steps consist of:

• Isolate the variable \(x\) by adding 5 to both sides to get \(x > 5\).
• This inequality means that \(x\) must be greater than 5 for the function to work properly.

Solving inequalities requires these steps because:

• The solution indicates the inputs that maintain the mathematical validity of the function.
• In this context, it helps us define where the logarithm is real and applicable.
• Lastly, it highlights any restrictions on the domain of the function, such as avoiding negative numbers inside the logarithm.

Understanding how to solve these inequalities allows for better comprehension of function behavior and limits.