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Chapter 1: Problem 24

Sketch a possible graph for a function \(f\) with the specified properties. (Many different solutions are possible.) $$ \begin{array}{l}{\text { (i) the domain of } f \text { is }(0,+\infty)} \\\ {\text { (ii) } f(1)=0} \\ {\text { (iii) the } y \text { -axis is a vertical asymptote for the graph of } f} \\ {\text { (iv) } f(x)<0 \text { if } 0

Short Answer

Expert verified
Sketch a curve that approaches the \( y \)-axis from the right (asymptote), passes through \( (1,0) \), lies below the \( x \)-axis for \( 0 < x < 1 \), and rises or levels off after \( x=1 \).

Step by step solution

01

Understand Function Properties

Begin by analyzing the properties given for the function \( f \). The properties include: (i) the domain is \( (0, +\infty) \), (ii) \( f(1) = 0 \), (iii) the \( y \)-axis is a vertical asymptote, and (iv) \( f(x) < 0 \) for \( 0 < x < 1 \). Sketching a graph involves understanding how these properties influence the shape and behavior of the graph.
02

Sketch the Domain

According to property (i), the domain of \( f \) is \((0, +\infty)\). This means that the function is not defined at \( x = 0 \) and can take values for any positive \( x \). In your graph, start plotting lines or curves beginning to the right of the \( y \)-axis (immediately after \( x = 0 \)), extending indefinitely to the right.
03

Plot the Point \( f(1)=0 \)

Property (ii) states \( f(1) = 0 \), which means the function intersects the \( x \)-axis at \( x = 1 \). On your graph, make a point at \((1,0)\). This will be a key feature of the graph where it touches or crosses the \( x \)-axis.
04

Draw the Vertical Asymptote

Property (iii) states that the \( y \)-axis (\( x = 0 \)) is a vertical asymptote for the graph. A vertical asymptote means the function will approach but never actually touch or cross the \( y \)-axis. Draw a dashed line at \( x = 0 \) to represent this asymptote, indicating that the function gets infinitely close to this line but does not reach it as \( x \) approaches zero.
05

Determine Function Behavior in \( 0

According to property (iv), \( f(x) < 0 \) when \( 0 < x < 1 \). This means the graph lies below the \( x \)-axis between \( x = 0 \) and \( x = 1 \). Sketch a curve starting near \( x = 0 \) (left of \( x = 1 \)) that is below the \( x \)-axis, and gradually slopes upward to meet the point \((1,0)\).
06

Sketch the Curve After \( x=1 \)

Since the domain extends beyond \( x = 1 \), continue the graph from the point \( (1,0) \). This part does not have specific properties given, allowing creative freedom. One possibility is to make the function rise gradually, suggesting it is approaching a constant positive value, or simply asymptotically approaching a horizontal line at some \( y > 0 \).
07

Finalizing the Graph

Ensure that the entire graph is consistent with the given properties: (i) it exists for \( x > 0 \); (ii) intersects at \( f(1) = 0 \); (iii) avoids the \( y \)-axis due to the asymptote; and (iv) is below the \( x \)-axis between \( x = 0 \) and \( x = 1 \). Evaluate if the sketched curve reflects these criteria and make adjustments accordingly.

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Key Concepts

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

Vertical Asymptote
A vertical asymptote is a line on the graph where the function value approaches infinity as the input gets closer to a certain value. In this context, the line is the y-axis, or the vertical line at \( x = 0 \). This means that as \( x \) approaches zero from the right, the function \( f(x) \) either increases or decreases without bound, but never actually touches or crosses this line.
To sketch a vertical asymptote, draw a dashed line on the graph to symbolize it, and remember that the behavior of \( f(x) \) near \( x = 0 \) will inform how it behaves along this line. This provides a boundary-like effect for the function's behavior as you sketch the rest of the graph.
Function Domain
The domain of a function defines all the possible \( x \) values for which the function is defined. For the function \( f \), the domain is \( (0, +\infty) \), indicating the graph only includes values greater than zero.
This means the function does not exist for any \( x \leq 0 \). When sketching the graph, commence plotting directly to the right of the y-axis (at \( x = 0 \)) and continue indefinitely to the right.
Be mindful that any points or sections of the graph shouldn't extend or touch \( x \leq 0 \), ensuring the graph adheres to the domain constraint.
X-Intercept
The x-intercept is the point where the graph crosses or touches the x-axis. This is crucial as it provides a fixed reference point in graph sketching. According to the given properties, \( f(1) = 0 \) is the x-intercept.
To sketch this, plot a point at \( (1, 0) \). This can act like a 'hinge' for your graph, affecting how the curve behaves as it approaches and leaves this point. Consider this x-intercept as a checkpoint ensuring your graph comprehensively meets the criteria specified for \( f(x) \).
Function Behavior
The behavior of the function \( f \) between \( 0In this range, the graph should be sketched below the x-axis. Begin near \( x = 0 \) 鈥� remember, the domain starts here 鈥� and end at the x-intercept point \( (1,0) \). The curve should start near the vertical asymptote, remain below the x-axis, then slope upwards reaching the x-intercept exactly when \( x = 1 \).
  • Ensure consistent behavior: maintain the below-x-axis position as you approach \( x = 1 \).
  • Use the x-intercept as a meeting point for this decrease, adhering to the conditions outlined.
After \( x = 1 \), there's freedom in describing the graph's behavior, such as continuing to rise or finding a horizontal asymptote.

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Most popular questions from this chapter

Find the discontinuities, if any. $$ f(x)=\sqrt{2+\tan ^{2} x} $$

The population \(p\) of the United States (in millions) in year \(t\) may be modeled by the function $$ p=\frac{50371.7}{151.3+181.626 e^{-0.031636(t-1950)}} $$ (a) Based on this model, what was the U.S. population in \(1950 ?\) (b) Plot \(p\) versus \(t\) for the 200 -year period from 1950 to 2150 . (c) By evaluating an appropriate limit, show that the graph of \(p\) versus \(t\) has a horizontal asymptote \(p=c\) for an appropriate constant \(c\). (d) What is the significance of the constant \(c\) in part (b) for population predicted by this model?

The notion of an asymptote can be extended to include curves as well as lines. Specifically, we say that curves \(y=f(x)\) and \(y=g(x)\) are asymptotic as \(x \rightarrow+\infty\) provided $$ \lim _{x \rightarrow+\infty}[f(x)-g(x)]=0 $$ In these exercises, determine a simpler function \(g(x)\) such that \(y=f(x)\) is asymptotic to \(y=g(x)\) as \(x \rightarrow+\infty\) or \(x \rightarrow-\infty\) Use a graphing utility to generate the graphs of \(y=f(x)\) and \(y=g(x)\) and identify all vertical asymptotes. $$ f(x)=\frac{x^{2}-2}{x-2} $$

The notion of an asymptote can be extended to include curves as well as lines. Specifically, we say that curves \(y=f(x)\) and \(y=g(x)\) are asymptotic as \(x \rightarrow+\infty\) provided $$ \lim _{x \rightarrow+\infty}[f(x)-g(x)]=0 $$ In these exercises, determine a simpler function \(g(x)\) such that \(y=f(x)\) is asymptotic to \(y=g(x)\) as \(x \rightarrow+\infty\) or \(x \rightarrow-\infty\) Use a graphing utility to generate the graphs of \(y=f(x)\) and \(y=g(x)\) and identify all vertical asymptotes. $$ f(x)=\frac{-x^{3}+3 x^{2}+x-1}{x-3} $$

Find the limits. $$ \lim _{h \rightarrow 0} \frac{1-\cos 3 h}{\cos ^{2} 5 h-1} $$
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