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Chapter 4: Problem 75

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

Short Answer

Expert verified
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 by step solution

01

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 \(x < -1\).
02

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.
03

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.
04

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).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Women's Heights The distribution of heights of American women (between 30 and 39 years of age) can be approximated by the function \(p=0.163 e^{-(x-64.9)^{2} / 12.03}, \quad 60 \leq x \leq 74\) where \(x\) is the height (in inches) and \(p\) is the percent (in decimal form). Use a graphing utility to graph the function. Then determine the average height of women in this age bracket. (Source: U.S. National Center for Health Statistics)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(3 \ln 5 x=10\)

Find the time required for a \(\$ 1000\) investment to double at interest rate \(r\), compounded continuously.\(r=0.0725\)

(a) \(I=10^{-3}\) watt per square meter (loud car horn) (b) \(I \approx 10^{0}\) watt per square meter (threshold of pain)

Population The populations \(P\) of the United States (in thousands) from 1990 to 2005 are shown in the table. (Source: U.S. Census Bureau)$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Population } \\ \hline 1990 & 250,132 \\ \hline 1991 & 253,493 \\ \hline 1992 & 256,894 \\ \hline 1993 & 260,255 \\ \hline 1994 & 263,436 \\ \hline 1995 & 266,557 \\ \hline 1996 & 269,667 \\ \hline 1997 & 272,912 \\ \hline \end{array} $$$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Population } \\ \hline 1998 & 276,115 \\ \hline 1999 & 279,295 \\ \hline 2000 & 282,403 \\ \hline 2001 & 285,335 \\ \hline 2002 & 288,216 \\ \hline 2003 & 291,089 \\ \hline 2004 & 293,908 \\ \hline 2005 & 296,639 \\ \hline \end{array} $$(a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=0\) corresponding to 1990 . (b) Use the regression feature of a graphing utility to find an exponential model for the data. Use the Inverse Property \(b=e^{\ln b}\) to rewrite the model as an exponential model in base \(e\). (c) Use the regression feature of a graphing utility to find a linear model and a quadratic model for the data. (d) Use a graphing utility to graph the exponential model in base \(e\) and the models in part (c) with the scatter plot. (e) Use each model to predict the populations in 2008 , 2009 , and 2010 . Do all models give reasonable predictions? Explain.
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