/*! 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 43 Graph \(f(x)=4^{x}\) and \(g(x)=... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 43

Graph \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\) in the same rectangular coordinate system.

Short Answer

Expert verified
The graphs of \(f(x)=4^x\) and \(g(x)=log_4 x\) are reflections of each other across the line \(y=x\) due to the property of exponential and logarithmic functions being inverses. The graph of \(f(x)=4^x\) passes through \((0,1)\), \((1,4)\), and \((-1, 0.25)\) while the graph of \(g(x)=log_4 x\) has a vertical asymptote at \(x=0\) and passes through \((1,0)\) and \((4,1)\).

Step by step solution

01

Graph the exponential function

To draw \(f(x)=4^x\), remember that exponential functions of the form \(f(x)=a^x\) always pass through the point \((0, 1)\). Since our base is 4, the function will rise rapidly. For \(x=0\), \(y=1\); for \(x=1\), \(y=4\); for \(x=-1\), \(y=0.25\). Plot these values on a graph.
02

Graph the logarithmic function

To draw \(g(x)=log_4 x\), remember that logarithmic functions of the form \(g(x)=log_a x\) have a vertical asymptote at \(x=0\), which implies that the function is undefined for \(x \leq 0\). For \(x=1\), \(y=0\); for \(x=4\), \(y=1\). Plot these points on the same graph where you plotted \(f(x)=4^x\).
03

Identify and explain symmetry

Observe that the two graphs are reflections of each other across the line \(y=x\). This is a consequence of the logarithm and exponential functions being inverses of one another.

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

Given \(f(x)=\frac{2}{x+1}\) and \(g(x)=\frac{1}{x},\) find each of the following: a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) (Section \(2.6, \text { Example } 6)\)

Use the proof of the product rule in the appendix to prove the quotient rule.

Write as a single term that does not contain a logarithm: $$ e^{\ln 8 x^{5}-\ln 2 x^{2}} $$

Explaining the Concepts Describe a difference between exponential growth and logistic growth.

By 2019 , nearly \(\$$ I out of every \)\$ 5\( spent in the U.S. economy is projected to go for health care. The bar graph shows the percentage of the U.S. gross domestic product \)(G D P)\( going toward health care from 2007 through \)2014,\( with a projection for 2019 The data can be modeled by the function \)f(x)=1.2 \ln x+15.7\( where \)f(x)\( is the percentage of the U.S. gross domestic product going toward health care \)x\( years after \)2006 .\( Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \)2009 .\( Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \)18.5 \%$ of the U.S. gross domestic product go toward health care? Round to the nearest year.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.