/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 269 For the following exercises, ske... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 269

For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. $$ f(x)=\ln (x+1) $$

Short Answer

Expert verified
Domain: \((-1, \infty)\), Range: \((-\infty, \infty)\), Vertical Asymptote: \(x=-1\).

Step by step solution

01

Identify the Parent Function

The given function is a natural logarithm function, represented as \( f(x) = \ln(x+1) \). The parent function for a natural logarithm is \( f(x) = \ln x \). This information helps in understanding how the shift affects the graph.
02

Determine Horizontal Shift

The function \( \ln(x+1) \) indicates a horizontal shift. For the two logarithmic expressions \( f(x) = \ln(x) \) and \( f(x) = \ln(x+1) \), the term \(+1\) inside the logarithm implies the graph is shifted 1 unit left on the x-axis.
03

Identify the Domain of the Funct ion

For the function \( \ln(x+1) \), we need \( x+1 > 0 \) for the logarithm to be defined, which simplifies to \( x > -1 \). Hence, the domain of the function is \( x \in (-1, \infty) \).
04

Identify the Range of the Function

The range of a logarithmic function is all real numbers, \( (-\infty, \infty) \), because as \( x \) approaches infinity, the log function increases without bound, and as \( x \) approaches -1 from the right, the log function approaches negative infinity.
05

Determine the Vertical Asymptote

Vertical asymptotes for logarithmic functions occur where the argument of the logarithm is zero. Solving \( x+1=0 \) gives \( x = -1 \) as the point for the vertical asymptote.
06

Sketch the Graph

Begin by drawing the vertical asymptote line at \( x = -1 \). Mark the domain \( x > -1 \). The graph starts near negative infinity as \( x \) approaches -1 from the right, and it rises gradually without bound as \( x \) increases. Sketch this curve carefully, reflecting the characteristics of the natural logarithm with the specified horizontal shift.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Natural Logarithm
The natural logarithm is a specifically significant logarithmic function denoted as \( \ln(x) \). Unlike other logarithms, it has a base of \( e \), an irrational constant approximately equal to 2.718. Natural logarithms are widely used due to their natural properties, mostly in calculus and natural growth models.
  • They convert multiplication into addition, simplifying calculations involving exponential functions.
  • In the function \( f(x) = \ln(x) \), \( x \) must be greater than zero, as log of zero or negative numbers is undefined.
Compared to standard logarithms, the natural logarithm is vital in deriving many mathematical properties and equations due to its seamless interaction with exponential functions.
Horizontal Shift
When you encounter expressions like \( f(x) = \ln(x+1) \), it signals a horizontal shift of the graph on the x-axis. Understanding Horizontal Shifts:
  • If \( x \) is replaced by \( x + c \), where \( c \) is a constant, the graph shifts \( c \) units to the left.
  • If \( x \) is replaced by \( x - c \), the graph shifts \( c \) units to the right.
For our function \( \ln(x+1) \), the graph is shifted one unit to the left compared to its parent function \( \ln(x) \). This shift does not alter the shape of the graph; it only affects its position on the x-axis.
Domain and Range
Determining the domain and range of a function helps in understanding where the function is defined and what values it can take. Domain: For the function \( f(x) = \ln(x+1) \), the expression inside the logarithm, \( x+1 \), must be greater than zero. This translates to:
  • Solving the inequality \( x+1 > 0 \) gives \( x > -1 \).
  • Therefore, the domain is \( x \in (-1, \infty) \).
Range: The range of a logarithmic function is not restricted and spans all real numbers:
  • As \( x \) approaches infinity, \( \ln(x+1) \) increases without bound.
  • As it approaches the boundary of the domain from the right, the log value approaches negative infinity.
  • Consequently, the range is \( (-\infty, \infty) \).
Vertical Asymptote
A vertical asymptote is a vertical line that indicates where a function鈥檚 graph might tend towards infinity, but never actually touches or crosses. In our logarithmic function \( \ln(x+1) \), determining the vertical asymptote involves finding when \( x+1 \) equals zero. Identifying Vertical Asymptote:
  • Set the expression inside the logarithm equal to zero: \( x+1=0 \).
  • This gives \( x = -1 \) as the equation of the vertical asymptote.
This asymptote represents the boundary the function's graph approaches as \( x \) nears \(-1\) from the right, spiraling towards negative infinity. Graphically, it鈥檚 crucial to draw this line to accurately reflect the behavior of the natural logarithm.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The function \(H(t)=8 \sin \left(\frac{\pi}{6} t\right)\) models the height \(H(\text { in feet })\) of the tide \(t\) hours after midnight. Assume that \(t=0\) is midnight. a. Find the amplitude and period. b. Graph the function over one period. c. What is the height of the tide at 4:30 a.m.?

Solve the following trigonometric equations on the interval \(\theta=[-2 \pi, 2 \pi]\) exactly. $$ \sec ^{2} x-2 \sec x+1=0 $$

For the following problems, state the domain and range of the given functions: \(f=x^{2}+2 x-3, \qquad g=\ln (x-5), \qquad h=\frac{1}{x+4}\) $$ h $$

For the following exercises, solve the exponential equation exactly. $$ e^{3 x}-15=0 $$

Find the degree, \(y\) -intercept, and zeros for the following polynomial functions. $$ f(x)=x^{3}+2 x^{2}-2 x $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.