Problem 64 - Precalculus : Building Concepts and Connections

Question

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=\log _{5}(x-2)$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\log_{5}(x-2)\) is \(x>2\), and the function has a vertical asymptote at \(x=2\). The graph of the function starts from \(x=2\) and then increases as \(x\) increases.

Step 1: Identify the Domain

Set the argument of the logarithm, \(x-2\), greater than zero and solve for \(x\). This is because logarithms are only defined for positive numbers. Solving this inequality: \(x-2>0\) gives, \(x>2\). So, the domain of this function is \(x>2\).

Step 2: Identify the Asymptotes

In general, the graph of a logarithmic function has a vertical asymptote at the x-value where the argument of the logarithm equals zero. So, in this case, the vertical asymptote of the function \(f(x)=\log_{5}(x-2)\) is at \(x=2\).

Step 3: Graph the Function

To graph the function, start by drawing the vertical asymptote at x=2. Then, plot a few points for \(x>2\) and connect them to show the curve of the graph. The function will start from the point where \(x=2.0\) and then, will increase as \(x\) increases.

Key Concepts

Domain of a Function

When determining the domain of a logarithmic function like \( f(x) = \log_{5}(x-2) \), it is essential to remember that the logarithm is only defined for positive arguments. This means we must solve the inequality \( x-2 > 0 \). Solving this gives \( x > 2 \).
Hence, the domain of the function is all real numbers greater than 2.
This implies that the function will not accept any input less than or equal to 2, thus the function's graph will only exist for these values.
Understanding the domain helps us establish the limits within which the function behaves regularly and predicts its behavior accurately on a graph.

Asymptotes

In the context of logarithmic functions, a vertical asymptote is a line on a graph where the function approaches but never actually reaches or crosses.
For the function \( f(x)=\log_{5}(x-2) \), the vertical asymptote occurs at the point where the argument of the log is zero.
Solving \( x-2=0 \) gives \( x=2 \). Thus, \( x=2 \) is the vertical asymptote for this function.
The graph will get closer and closer to this line as \( x \) approaches 2 from the right, but it will never actually touch or intersect it.
It's crucial to pay attention to asymptotes when graphing, as they dictate the behavior and the limitation points of the function.

Graphing Functions

Graphing a logarithmic function involves understanding its domain and identifying certain key features, like vertical asymptotes, which guide the shape of the graph.
For \( f(x) = \log_{5}(x-2) \), we start by sketching the vertical asymptote at \( x = 2 \).
Next, we choose several values of \( x \) that are greater than 2, and calculate the corresponding \( f(x) \) values to get the plot points.
As \( x \) increases, the value of the function also increases but at a decreasing rate.
Once these points are plotted, a smooth curve is drawn, starting from close to the asymptote and moving upwards to the right.
It is vital to depict the function's behavior clearly, reflecting the domain restrictions and asymptotic behavior accurately.