Problem 75 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The graph of \(f(x)=\log(x+1)\) decreases without limit as \(x\) approaches -1 from the right, pass through the point (0,0), and slowly increases as \(x\) goes to positive infinity. The key points of the graph are the vertical asymptote at \(x = -1\) and the x-intercept at (0,0).

Step 1: Understand the function

The function \(f(x)=\log(x+1)\) is a simple logarithmic function, which is translated one unit to the left. The base of the logarithm isn't specified, so it is understood to be base 10. This function will be undefined for \(x < -1\).

Step 2: Identify the key points of the graph

Analyzing the function, there are two important key points of the graph: a vertical asymptote at \(x = -1\), and the point (0,0) where the graph of the function intersects the x-axis.

Step 3: Use a graphing utility

Using a graphing utility, input the function and select a suitable viewing window to capture the essential features of the graph. Since there is no restriction given on the x-range, a standard window of (-10, 10) for \(x\) and (-10, 10) for \(f(x)\) is sufficient to show the behavior of the graph.

Step 4: Interpret the graph

The graph of the function \(f(x)=\log(x+1)\) should decrease without bound as \(x\) approaches -1 from the right, and should increase slowly as \(x\) goes to positive infinity. The graph should also pass through the point (0,0).

Key Concepts

Graphing Utilities

Graphing utilities are valuable tools in understanding the behavior of functions by providing a visual representation of their graphs. They help to quickly observe important characteristics such as trends, key points, and the general shape of a graph. For the logarithmic function \(f(x) = \log(x+1)\), using a graphing utility allows us to find:

• How the graph approaches the vertical asymptote
• The behavior of the graph as \(x\) approaches infinity
• The point where the graph crosses the x-axis

To graph \(f(x) = \log(x+1)\), simply input the function into the utility and select a suitable viewing window. A standard viewing window of \((-10, 10)\) for \(x\) and \((-10, 10)\) for \(f(x)\) often provides a complete view of the function's behavior across its domain. Adjusting the window can enhance specific features, like zooming in on the asymptote at \(x = -1\), or zooming out to observe asymptotic behavior towards infinity.

Vertical Asymptotes

Vertical asymptotes are imaginary lines where a function's value becomes unbounded as it approaches the line. In simpler terms, the function can shoot upwards to positive infinity or downwards to negative infinity near this asymptote. For the function \(f(x) = \log(x+1)\), the vertical asymptote occurs at \(x = -1\).As \(x\) approaches -1 from the right, the value of \(f(x)\) decreases boundlessly. This feature is a critical aspect of graphing the function, as it shows an essential boundary that the function never crosses. Hence, this asymptote is a guide to understanding the limits of the function as \(x\) changes.

Function Translation

Function translation involves shifting the graph of a function horizontally, vertically, or both without altering its shape. For the logarithmic function \(f(x) = \log(x+1)\), it is shifted 1 unit to the left compared to the basic logarithm function \(f(x) = \log(x)\). Function translation impacts how and where a graph appears on a coordinate system. With \(f(x) = \log(x+1)\), the graph starts shifting before it would naturally start in a basic log function. It affects:

• The position of the vertical asymptote, which moves from \(x = 0\) to \(x = -1\)
• The intercept with the x-axis, which appears at \(x = 0\) in this case

Understanding translation helps in adjusting graph viewing windows and better predicting the key features of the graph, enhancing comprehension of the function's behavior.