/*! 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 16 sketch the graph of each functio... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 16

sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=\ln (x-3) $$

Short Answer

Expert verified
The graph of \( y = \ln(x-3) \) has a vertical asymptote at \( x=3 \), an x-intercept at \( (4, 0) \), and increases without bound as \( x \to \infty \).

Step by step solution

01

Identify the domain

The function given is the natural logarithm: \( y = \ln(x-3) \). The expression \( x-3 \) inside the logarithm must be greater than zero because the logarithm is undefined for zero and negative numbers. Therefore, the domain of the function is \( x-3 > 0 \), which simplifies to \( x > 3 \). The domain of the function is all real numbers greater than 3.
02

Determine the vertical asymptote

A vertical asymptote occurs where the function approaches infinity. For \( y = \ln(x-3) \), as \( x \) approaches 3 from the right (i.e., values slightly greater than 3), \( y \) will tend towards negative infinity. Therefore, there is a vertical asymptote at \( x = 3 \).
03

Identify the x-intercept

To find the x-intercept, set \( y = 0 \) and solve for \( x \). \( 0 = \ln(x-3) \) implies that \( x-3 = e^0 = 1 \), thus \( x = 4 \). The x-intercept is at the point \( (4, 0) \).
04

Plot key points and sketch the graph

With the domain \( x > 3 \) and a vertical asymptote at \( x = 3 \), we know the curve starts right after \( x = 3 \) and descends from infinity as \( x \to 3^+ \). At \( x = 4 \), the point is \( (4,0) \). As \( x \to \infty \), \( y \to \infty \) as well. Sketch the graph starting from the vertical asymptote at \( x = 3 \), passing through \( (4,0) \), increasing to the right, and approaching infinity as \( x \to \infty \).
05

Analyze the behavior of the graph

As \( x \to 3^+ \), the graph plunges towards negative infinity due to the vertical asymptote. Post the x-intercept at \( (4, 0) \), the graph increases steadily without bound. The basic shape of \( \ln(x-3) \) closely mirrors that of \( \ln(x) \) with a horizontal shift to the right by 3 units.

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Key Concepts

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

Natural Logarithm Function
The natural logarithm function, denoted as \( \ln(x) \), is a logarithm to the base \( e \). The function \( y = \ln(x-3) \) represents a horizontal shift of the basic natural logarithm function by 3 units to the right.

Here are some key characteristics of the natural logarithm function:
  • Its domain includes all positive real numbers since logarithms are defined only for positive values.
  • The range is all real numbers because the output can be any real value as \( x \) varies.
  • The graph of \( \ln(x) \) approaches negative infinity as \( x \) approaches zero from the right.
Understanding these properties helps in sketching graphs of transformed logarithmic functions like \( y = \ln(x-3) \). In this function, any point \( x \) in the basic function's domain is shifted by 3 units.
Graphing Without Calculator
When graphing functions like \( y = \ln(x-3) \) without a calculator, it's crucial to use key features of the function's equation. Begin by identifying transformations and characteristics like domain, x-intercepts, and asymptotes.

Use these features to plot:
  • Domain: Identify values of \( x \) where the function exists. For \( \ln(x-3) \), it's \( x > 3 \).
  • X-Intercept: Calculate where the function crosses the x-axis.
  • Asymptotes: Find lines the graph will approach but never touch.
With \( y = \ln(x-3) \), mark the x-intercept at \( (4, 0) \), and the vertical asymptote at \( x = 3 \). Sketch the graph starting after \( x = 3 \), decreasing toward negative infinity as it approaches the asymptote, and increasing towards positive infinity as \( x \) goes to infinity.
Function Domain Analysis
The domain of any function is key to understanding where it is defined. For \( y = \ln(x-3) \), analyze the expression inside the logarithm, \( x-3 \).
To ensure this expression is positive, solve the inequality:
  • \( x-3 > 0 \)
  • \( x > 3 \)
Thus, the domain of the function \( y = \ln(x-3) \) is all real numbers greater than 3. This means the function does not exist for any \( x \leq 3 \), as logarithms of non-positive numbers are undefined.
Understanding the domain helps to determine the behavior of the function and informs the sketching of the graph.
Vertical Asymptote
A vertical asymptote is a line that a graph approaches but never actually meets. For logarithmic functions, vertical asymptotes occur where the argument of the logarithm approaches zero.

In the function \( y = \ln(x-3) \), a vertical asymptote occurs at \( x = 3 \) because:
  • As \( x \to 3^+ \), \( \ln(x-3) \to -\infty \).
  • This behavior shows that as you get closer to \( x = 3 \) from the right, the function value plunges toward negative infinity.
This insight is crucial in accurately sketching the function's graph and understanding how the function behaves near this asymptote.
X-Intercept
An x-intercept is a point where the graph crosses the x-axis, which occurs where the function value equals zero. To find the x-intercept of \( y = \ln(x-3) \), solve:

\[ 0 = \ln(x-3) \]
This implies that \( x-3 = 1 \), because \( e^0 = 1 \). Solving this gives:
  • \( x = 4 \)
The x-intercept is at the point \( (4, 0) \). This information helps plot the graph accurately, as it identifies where the function changes sign and crosses the x-axis.

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