Problem 106 - College Algebra with Modeling and Visualization

Question

Graph \(f\) and state its domain. $$f(x)=\log (x-3)$$

Step-by-Step Solution

Answer

Domain: \((3, \infty)\). Graph: Vertical asymptote at \(x = 3\), increasing rightward.

Step 1: Identify the Function and Its Type

The given function is \(f(x) = \log(x-3)\). This is a logarithmic function. Logarithmic functions are defined only for positive arguments, which means the expression inside the logarithm should be greater than zero.

Step 2: Find the Domain of the Function

To find the domain, set the expression inside the logarithm greater than zero: \(x-3 > 0\). Solving this inequality gives \(x > 3\). Therefore, the domain of \(f(x)\) is all real numbers greater than 3, which is represented as \((3, \infty)\) in interval notation.

Step 3: Graph the Function

To graph \(f(x) = \log(x-3)\), start your graph at \(x = 3\), moving rightward because the domain is \(x > 3\). As \(x\) approaches 3 from the right, the function approaches negative infinity, denoting a vertical asymptote at \(x = 3\). As \(x\) increases, \(f(x)\) increases slowly without bound.

Step 4: Analyze Important Points

Choose values for \(x\) greater than 3 to find key points. For example, at \(x = 4\), \(f(4) = \log(4-3) = \log(1) = 0\). At \(x = 5\), \(f(5) = \log(5-3) = \log(2)\), which is approximately 0.301. These points help in visualizing the graph.

Key Concepts

Domain of a Function

The domain of a function defines the set of all possible input values (usually designated by "x") for which the function is defined. In the case of logarithmic functions like \(f(x) = \log(x-3)\), the domain is crucial to ensure the expression inside the logarithm is positive.
To determine this, set the argument of the logarithm to be greater than zero: \(x-3 > 0\). Solving this inequality, you find \(x > 3\). Hence, the domain of \(f\) is all real numbers greater than 3.
In interval notation, this is expressed as \((3, \infty)\).

• Always remember: you can't take the logarithm of zero or negative numbers.
• Logarithmic functions only exist when their arguments are positive.

This emphasizes how understanding the domain is foundational in analyzing logarithmic functions.

Graphing Functions

Graphing functions allows us to visualize their behavior and understand their properties better. For the function \(f(x) = \log(x-3)\), the graph starts at \(x = 3\). This is because the domain begins at values greater than 3.
A key feature of this graph is the vertical asymptote at \(x = 3\). Here, as \(x\) approaches 3 from the right (but doesn't reach 3), the function \(f(x)\) tends towards negative infinity.

• Select key points within the domain to help plot the graph. For example, at \(x = 4\), \(f(x) = 0\). This point passes through the x-axis.
• As \(x\) increases, \(f(x)\) grows gradually, increasing without bound.

Graphing helps identify how the function behaves near its vertical asymptote and how it trends over greater values of \(x\).

Mathematical Analysis

Mathematical analysis in the context of functions involves understanding their behavior, limits, and possible applications. In \(f(x) = \log(x-3)\), the analysis explains how the logarithmic function behaves as \(x\) changes within its domain.

• As \(x\) approaches 3, the function value decreases dramatically, showing a vertical asymptote at \(x = 3\).
• Past the asymptote, each incremental increase in \(x\) results in a slow increase in \(f(x)\).

This "slow growth" is a typical characteristic of logarithmic functions. They grow infinitely, but at a decreasing rate as \(x\) grows. This property is valuable in mathematical analysis, particularly in fields like computer science and biology, where exponential growth eventually slows.