Problem 77 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window.\(f(x)=\ln (x-1)\)

Step-by-Step Solution

Answer

For the function \(f(x)=\ln (x-1)\), the domain is \(x>1\) as \(x-1\) should be positive. Initially, you can take the viewing window for \(x\) as [1,5] and for \(y\) as [-10,10]. If this window doesn't show a clear graph, continue to modify it until you get a clear visualization of the function.

Step 1: Understanding the Function

The given function is \(f(x)=\ln (x-1)\). It is a logarithmic function where the base is the number e (approximately equal to 2.718). Since it's a logarithmic function, it means the domain for \(x\) should be greater than 1 because the term inside the logarithm (the argument) must be positive.

Step 2: Identify the Viewing Window

The domain of the function is \(x>1\), which defines the x-values on the graph. For the y-values, since logarithmic functions can have any real number as their output, the range is often set to cover typical viewing windows such as \(-10\) to \(10\). So, an initial viewing window could be [1,5] for \(x\) and [-10,10] for \(y\) values.

Step 3: Graph the Function

Input the function \(f(x)=\ln (x-1)\) into the graphing utility. Make sure to set the viewing window as identified in the previous step. The graph should resemble a logarithmic graph that goes towards positive infinity as x increases and towards negative infinity as x approaches 1 from the right.

Step 4: Adjust if Necessary

If the graph doesn't show the characteristics of the function clearly in the initial viewing window, adjust the viewing window. Gradually expand the x and y values range until the function behavior is clearly visible.

Key Concepts

Domain and Range

Every function has a domain and a range. For logarithmic functions like \(f(x)=\ln (x-1)\), understanding these concepts is crucial, especially because they affect how the graph will appear.

• Domain: This refers to all the possible input values \(x\) for the function. For a logarithmic function \(\ln(x-1)\), the input \(x-1\) must be positive. Therefore, \(x\) must be greater than 1. This means the domain of the function is \(x > 1\).
• Range: This is all the possible output values. For a natural logarithmic function, the range is all real numbers, meaning the graph can stretch from negative to positive infinity along the y-axis. This freedom in the y-value is what gives the function its characteristic curve.

Knowing the domain and range is vital to set the right viewing window when graphing, ensuring that the essential parts of the graph are visible.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, and they have unique characteristics.

• The basic form is \(f(x) = \log_b(x)\), where \(b\) is the base of the logarithm.
• In \(f(x) = \ln(x-1)\), the base \(e\) is approximately 2.718, and the \(x-1\) suggests a horizontal shift in the function.

Logarithmic functions typically increase slowly: they have a steep rise near their vertical asymptote (a line the graph never touches), and become flatter as \(x\) grows.

• Vertical asymptote: For \(\ln(x-1)\), there is a vertical line at \(x = 1\), which the graph will never cross but will get infinitely close to.
• They have a natural domain that starts where the argument of the logarithm is greater than zero, in this case \(x > 1\).

Recognizing these features ensures accurate graphing and interpretation of the function's behavior.

Graphing Utilities

Graphing utilities are powerful tools that help visualize mathematical functions like \(f(x) = \ln(x-1)\), especially when handling complex or less intuitive graphs.

Following these steps ensures accurate graph results:

• Inputting the Function: Enter \(f(x) = \ln(x-1)\) into your graphing device.
• Setting the Viewing Window: Given the domain \(x > 1\), start with an x-range slightly above 1, like [1, 5]. Check the range with typical y-values like [-10, 10] to capture the curve's behavior.
• Adjusting the Window: If the initial view doesn't display the function's key characteristics well, adjust it. Gradually expand the x and y values until the graph is clear and visible.

A well-positioned window gives an insightful display of how the logarithmic function behaves over different inputs. This makes it easier to analyze trends and understand its mathematical properties.