/*! 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 35 Sketch the graph of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 35

Sketch the graph of the function and state its domain. $$ f(x)=\ln (x-1) $$

Short Answer

Expert verified
The graph of the function \(f(x) = \ln (x - 1)\) starts at \(x = 1\) with a vertical asymptote and increases without bound as \(x\) goes towards +infinity. The domain of the function is \(x > 1\).

Step by step solution

01

Understand The Base Function

The natural logarithm function, \(\ln (x)\), has the graph of y intercept at 0 and increasing without bound as \(x\) approaches +infinity, it is undefined for \(x 鈮� 0\), hence it has a vertical asymptote at \(x = 0\). When the function is \(\ln (x - 1)\), the entire graph of \(\ln (x)\) is shifted one unit to the right. This is because the \(x - 1\) in the argument of the logarithm delays the input into the logarithm function by 1.
02

Sketch The Graph

The graph of the function \(f(x) = \ln (x - 1)\) can be drawn by starting with the graph of \(y = \ln (x)\), and shifting all points one unit to the right. There will be a vertical asymptote at \(x = 1\), and the function increases without bound as \(x\) goes towards +infinity.
03

State The Domain

The domain of the function \(f(x) = \ln (x - 1)\) can be decided by considering when the function is defined. The natural logarithm is only defined for positive numbers, which means that \(x - 1 > 0\) for the function \(f(x) = \ln (x - 1)\). Solving this inequality gives \(x > 1\). Therefore, the domain of the function is \(x > 1\).

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!

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

In Exercises \(35-38\), use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l} f(x)=\frac{1}{x^{2}-25} \\ \lim _{x \rightarrow 5^{-}} f(x) \end{array} $$

Determine conditions on the constants \(a, b,\) and \(c\) such that the graph of \(f(x)=\frac{a x+b}{c x-a}\) is symmetric about the line \(y=x\).

Write the expression in algebraic form. \(\sin (\arccos x)\)

In Exercises \(25-34,\) find the limit. $$ \lim _{x \rightarrow(\pi / 2)} \ln |\cos x| $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \arcsin ^{2} x+\arccos ^{2} x=1 $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.