/*! 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 40 Find the domain, \(x\) -intercep... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 40

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$h(x)=\log _{4}(x-3)$$

Short Answer

Expert verified
The domain of \(h(x)\) is \(x>3\). The \(x\)-intercept is \(x = 4\). The vertical asymptote is \(x = 3\).

Step by step solution

01

Find the Domain

The domain of a log function \(h(x) = \log_b (x-a)\) is when \(x-a > 0\). So for the function \(h(x) = \log_4 (x-3)\), the domain would be \(x>3\). This is because log functions only take positive arguments.
02

Solve for the x intercept

Set \(h(x) = 0\) to solve for \(x\). We get \(0 = \log_4 (x-3)\). By definition of a logarithm, this implies that \(x-3 = 4^0\), since any number raised to the power of zero is 1. Therefore, \(x = 3+1 = 4\). Thus, the x-intercept is \(x = 4\).
03

Identify the Vertical Asymptote

For the function \(h(x) = \log_4 (x-3)\), the vertical asymptote is \(x = 3\). This is the boundary of the domain where the function tends to \(\pm \infty\).
04

Sketch the graph

Base the sketch on these findings. It should show the vertical asymptote at \(x = 3\), it should go through the x-intercept at \(x=4\), and the part of the graph should be to the right of \(x = 3\).

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

A logistic curve is also called a ________ curve.

Carbon 14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree growing today. A piece of ancient charcoal contains only \(15 \%\) as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal if the half-life of \({ }^{14} \mathrm{C}\) is 5715 years?

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1+\ln x}{2}=0$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1-\ln x}{x^{2}}=0$$

The demand equation for a limited edition coin set is \(p=1000\left(1-\frac{5}{5+e^{-0.001 x}}\right)\) Find the demand \(x\) for a price of (a) \(p=\$ 139.50\) and (b) \(p=\$ 99.99\).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.