Question

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window.\(f(x)=3 \ln x-1\)

Short answer

The function \(f(x)=3 \ln x-1\) has a graph which starts from the point \((1,-1)\) and increase more steeply compared to the graph of \(\ln x\), because of the vertical stretch by a factor of 3 and the whole graph shifts down by 1 unit.

Step-by-step solution

Identify the basic function

The function given is a logarithmic function, more specifically, a natural logarithm function. Basic form is \(\ln x\) which graph increases from left to right on the x axis. Here, the constant 3 and -1 are modifying the basic logarithm function.

Understand the modifications of the function

The given function \(f(x)=3 \ln x-1\) is a transformation of the basic log function. The 3 which multiply the \(\ln x\) will make graph stretch vertically by a factor of 3 and -1 which subtracts from 3\(\ln x\) will shift the graph down by 1 unit.

Determine an appropriate window

For logarithmic functions, the x values must be greater than 0. So, start with x values from 1 and view the y values within a reasonable range such that all the critical points and overall shape of the function can be determined.

Graph the function

Now, input the function into the graphing utility, make sure to set appropriate viewing window as determined in step 3. The graph will start at \((1,-1)\) and increases vertically by 3 units compared to basic log graph.

Key concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a fundamental mathematical function that is widely used throughout various aspects of mathematics, especially in calculus, complex analysis, and certain applications involving growth and decay processes. It is the inverse operation of exponentiation to the base \( e \), where \( e \approx 2.71828\) is Euler's number, a constant that's ubiquitous in pure and applied mathematics. When we take the natural logarithm of a number, we're essentially asking, 'To what power do we need to raise \( e \)' to get this number?'

The graph of a natural logarithm function \( y = \ln x \) is characterized by its curve that passes through the point \( (1,0) \) and gradually increases without bound as \( x \) goes to infinity while never crossing the \( y \) axis. It approaches the \( y \) axis asymptotically as \( x \) decreases toward zero. The function is only defined for positive values of \( x \), so when graphing, we need to ensure our domain excludes nonpositive numbers.

Transformations of Functions

Transformations of functions are various operations that modify the graph of a basic function in a systematic way. When we apply transformations to the natural logarithm function, or any function for that matter, we change its graph's size, position, and orientation but retain its overall shape. Notable transformations include:

• Vertical Stretch/Compression: Multiplying the function by a factor greater than 1 stretches it vertically. Conversely, multiplying by a factor between 0 and 1 compresses it vertically.
• Vertical Shifting: Adding or subtracting a number from the function shifts its graph up or down respectively.
• Horizontal Stretch/Compression: Multiplying \( x \) by a factor affects the horizontal stretching or compression of the graph.
• Horizontal Shifting: Adding or subtracting a number to \( x \) inside the function shifts the graph left or right.
• Reflection: Multiplying the entire function by -1 reflects it over the \( x \) axis, and multiplying \( x \) by -1 reflects it over the \( y \) axis.

In the given exercise, \( f(x)=3 \ln x-1 \) demonstrates a vertical stretch by a factor of 3 and a downward shift by 1 unit. Understanding these transformations allows one to predict the adjustments to the basic log graph and aid in graphing the new function.

Graphing Utilities

Graphing utilities are digital tools designed to help visualize mathematical functions. These utilities range from simple online graphing calculators to complex software designed for scientific computing. When dealing with logarithmic functions such as the natural logarithm, it's important to choose a graphing utility that can handle the intricacies and undefined regions of these functions.

To effectively use a graphing utility:

• Enter the function correctly by using the built-in syntax for logarithms.
• Choose the appropriate scale and range for the axes to ensure that the significant features of the function are visible.
• Adjust the resolution or the quality of the graph to get a clear picture of the behavior of the function.
• Use zoom and trace features to examine specific parts of the graph or to determine exact coordinates of points of interest.

Correct usage of these utilities simplifies the process of graphing complex functions and enhances one's understanding by providing a visual representation of abstract mathematical concepts.

Determining Viewing Window

Determining the correct viewing window for graphing a function is crucial because it ensures that all the important characteristics of the graph are captured. Getting the window wrong might mean missing out on critical points, asymptotes, or on the true nature of the function's behavior.

Here are steps to find an appropriate viewing window:

• Identify the domain and range of the function. For instance, the natural logarithm is defined for \( x > 0 \).
• Consider key points like intercepts, peaks, troughs, and any asymptotes.
• Choose the \( x \) and \( y \) values that frame these features properly without including too much empty space, which can make the graph appear smaller than needed.
• If you’re dealing with transformations, adjust the window to account for shifts and stretches.

In the exercise, since we deal with \( f(x)=3 \ln x-1 \), it is suggested to start with \( x=1 \) as the function is undefined for \( x<=0 \), and to use \( y \) values that comfortably include the point \( (1,-1) \) followed by sufficient vertical space to allow for the vertical stretch and continue beyond the initial portion of the function.