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Chapter 1: Problem 18

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=-\ln (1-x) $$

Short Answer

Expert verified
The graph has a vertical asymptote at \( x=1 \), passes through (0, 0), and decreases as x approaches 1 from the left.

Step by step solution

01

Identify the Basic Form of the Function

The function given is \( y = -\ln(1-x) \). The basic form of this function is a logarithmic function, specifically a natural logarithm with an inverted sign and shifted argument due to \(1-x\). This affects the domain and vertical shift but retains the logarithm characteristics.
02

Determine the Domain

The natural logarithm function \( \ln(u) \) is defined only for \( u > 0 \). Therefore, we must solve the inequality \( 1-x > 0 \) to find the domain of the function. Solving gives: \( x < 1 \). Thus, the domain of the function is \( (-\infty, 1) \).
03

Find any Intercepts

To find the y-intercept, set \( x = 0 \): \( y = -\ln(1-0) = -\ln(1) = 0 \). Therefore, the y-intercept is \( (0, 0) \). There is no x-intercept since \( y = -\ln(1-x) \) never equals zero for any \( x \) other than zero.
04

Analyze Asymptotes and Behavior

As \( x \to 1^- \), \( y = -\ln(1-x) \rightarrow -\infty \), indicating a vertical asymptote at \( x = 1 \). Also, as \( x \to -\infty \), \( y = \infty \). This will affect the tail behavior of the graph.
05

Sketch the Graph

The graph is a reflection over the x-axis of the logarithm function \( \ln(1-x) \), which is inverted and shifted. Draw the vertical asymptote at \( x = 1 \), plot the intercept at \( (0, 0) \), and ensure the graph decreases sharply as \( x \to 1^- \) and increases without bound negatively as \( x \to -\infty \).

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Key Concepts

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

Domain of a Function
The domain of a function refers to the set of all possible input values (usually x-values) for which the function is defined. When considering the function \( y = -\ln(1-x) \), identifying the domain is crucial because the natural logarithm, \( \ln(u) \), is only defined when \( u > 0 \).

For this particular function, the inside of the logarithm, \( 1-x \), must be positive. Therefore, we set the inequality \( 1-x > 0 \) and solve for \( x \), which gives us \( x < 1 \). This means the domain consists of all real numbers less than 1, so the domain can be written as \( (-\infty, 1) \).

Understanding the domain allows us to know where the graph of the function will exist on the x-axis.
Natural Logarithm
The natural logarithm, represented by \( \ln(x) \), is a special type of logarithm where the base is the mathematical constant \( e \), approximately equal to 2.718. This logarithm is profound in mathematics because it naturally arises in many natural growth and decay processes.

The function \( y = -\ln(1-x) \) involves a natural logarithm with the "\( 1-x \)" adjustment. Let's break down why this matters:
  • The negative sign in front of \( \ln(x) \) means the curve will be reflected over the x-axis, resulting in a downward slope.
  • The shift from \( x \) to \( 1-x \) affects the domain and the positioning of the curve on the graph.
It's crucial to note these modifications as they determine the function's behavior across its domain.
Asymptotes
An asymptote is a line that a graph approaches but never quite reaches. They can guide us in understanding the behavior of a function as it extends towards specific points in its domain or beyond.

For the function \( y = -\ln(1-x) \), as \( x \) approaches 1 from the left (\( x \to 1^- \)), the term \( 1-x \) approaches zero, causing \( \ln(1-x) \) to tend towards \( -\infty \). The negative in the function causes this to become \( \infty \), so \( y = -\ln(1-x) \to -\infty \).

This behavior results in a vertical asymptote at \( x = 1 \). This means the graph will decline sharply downward as it nears this line and provides insight into how the graph behaves as \( x \) increases within its domain toward 1.

Recognizing asymptotes is key in shaping the function's graph and understanding its limits.

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Most popular questions from this chapter

The following table is based on a functional relationship between \(x\) and \(y\) that is either an exponential or a power function: \begin{tabular}{cc} \hline \(\boldsymbol{x}\) & \(\boldsymbol{y}\) \\ \hline\(-1\) & \(0.398\) \\ \(-0.5\) & \(1.26\) \\ 0 & 4 \\ \(0.5\) & \(12.68\) \\ 1 & \(40.18\) \\ \hline \end{tabular} Use an appropriate logarithmic transformation and a graph to decide whether the table comes from a power function or an exponential function, and find the functional relationship between \(x\) and \(y\).

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=-2 x^{2}-3 $$

Boulinier et al. (2001) studied the dynamics of forest birdcommunities. They found that the mean local extinction rate of area-sensitive species declined with mean forest patch size. whereas the mean extinction rate of non-area- sensitive species did not depend on mean forest size. In the same coordinate system, graph the mean extinction rate as a function of mean forest patch size for (a) an area-sensitive species and (b) a non-area-sensitive species.

73\. The following table is based on a functional relationship between \(x\) and \(y\) that is either an exponential or a power function: \begin{tabular}{rl} \hline \(\boldsymbol{x}\) & \(\boldsymbol{y}\) \\ \hline 1 & \(1.8\) \\ 2 & \(2.07\) \\ 4 & \(2.38\) \\ 10 & \(2.85\) \\ 20 & \(3.28\) \\ \hline \end{tabular}

Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ L(c)=1.7 \times 10^{2.3 c} $$
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