Problem 71 - Precalculus with Limits

Question

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window. \(f(x)=\ln x+8\)

Step-by-Step Solution

Answer

The graph should be increasing, showing a vertical asymptote at x=0 and a y-intercept at y=8 in an appropriate viewing window.

Step 1: Identify the basic form of the function

The provided function is a logarithmic form. It can be written as \(f(x) = \ln{x} + 8\). This logarithmic function consists of a base \(e\) and a shifted up by eight units from the standard natural logarithm function \(\ln{x}\).

Step 2: Define the viewing window

In logarithmic graphs, the domain is all positive numbers. For this function, since the graph is shifted 8 units up due to '8', a good range to observe the main properties of the graph could be from -1 to 10 for y-values. For x-values, we can set the x range to be from 0.1 to 3 because natural log of x tends to negative infinity when x approaches zero from the right and we don't want to miss this important characteristic of the graph.

Step 3: Graph the function

Once the viewing window is set, use a graphing utility to graph \(f(x) = \ln{x} + 8\). Please observe the graph's behavior. The graph should have a vertical asymptote at x=0, a y-intercept at y=8 when x=1 because \( \ln{1} = 0\), and the graph should be increasing.

Key Concepts

Natural Logarithm

In mathematics, the natural logarithm is a logarithmic function that has the number e (approx. 2.71828) as its base. It is denoted as \( \ln(x) \) and is widely used in various fields such as science, engineering, and finance due to its unique properties.

The natural logarithm has a domain of all positive real numbers, as you cannot take the logarithm of zero or a negative number. When graphing \( \ln(x) \) on a coordinate plane, the function will produce a curve that increases slowly for larger values of \( x \) and moves toward negative infinity as it approaches zero. This behavior is characteristic of all logarithmic functions, but the natural logarithm describes this relation specifically for the constant \( e \) as the base.

Viewing Window

When graphing functions using a graphing utility or calculator, the viewing window is the area of the coordinate plane displayed on your screen. Selecting an appropriate viewing window is crucial to clearly observe the important characteristics and behavior of the function.

For logarithmic functions like \( f(x) = \ln{x} + 8 \), choosing a window that includes the function's y-intercept and shows the approach towards any asymptotes is essential. It is usually recommended to start with the standard window, then adjust the range and domain based on the specific function to capture its distinct features without losing detail or distorting the graph's representation.

Vertical Asymptote

A vertical asymptote is a vertical line that a function approaches but never actually reaches or crosses. Represented by an equation of the form \( x = a \), where \( a \) is a constant, the vertical asymptote is a boundary that the graph infinitely approaches but does not touch.

In the context of the natural logarithm function, the vertical asymptote occurs at \( x = 0 \). This means as \( x \) gets closer and closer to zero from the right, the function \( f(x) = \ln{x} + 8 \) heads towards negative infinity, indicating an undefined value at \( x = 0 \). This is a critical component to understand when graphing logarithmic functions because it informs the placement of the viewing window and the shape of the graph.

Graph Behavior

Understanding graph behavior helps to predict and analyze the function's path or trajectory on a coordinate plane. For the function \( f(x) = \ln{x} + 8 \), we expect certain behaviors:

• The graph will have a vertical asymptote at \( x = 0 \) and will never cross this boundary, depicting an undefined value at \( x = 0 \).
• The function's y-intercept is at \( f(1) = \ln{1} + 8 = 8 \) because the natural log of one is always zero.
• The graph will be always increasing since logarithmic functions are strictly increasing for all values in their domain.

These behaviors combined provide a clear mental image of the graph: it rises slowly to the right of the y-axis, increases more noticeably as \( x \) increases, and it shoots downwards as it nears the vertical asymptote at \( x = 0 \).