/*! 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 51 Graph functions \(f\) and \(g\) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 51

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=\left(\frac{1}{2}\right)^{x} \text { and } g(x)=\left(\frac{1}{2}\right)^{x-1}+1$$

Short Answer

Expert verified
The graphs of \(f(x) = 0.5^x\) and \(g(x) = 0.5^{x-1}+1\) decrease as \(x\) increases. The horizontal asymptote of \(f(x)\) is \(y = 0\) and the one for \(g(x)\) is \(y = 1\).

Step by step solution

01

Graph the function \(f(x)\)

Start by graphing the function \(f(x) = 0.5^x\). This is a simple exponential function where the base is \(0.5\). The graph will decrease as \(x\) increases and will never touch the x-axis, thus the x-axis (\(y = 0\)) is the horizontal asymptote.
02

Graph the function \(g(x)\)

Next, graph the function \(g(x) = 0.5^{x-1} + 1\). This function is similar to the first one, but it is shifted to the right by 1 unit and up by 1 unit. The left shift is due to the \(x-1\) in the exponent and the upward shift is due to the \(+1\) at the end of the function. The graph will decrease as \(x\) increases and will never touch the line \(y=1\), thus the line \(y = 1\) is the horizontal asymptote.
03

Identify the asymptotes

From the graphical representations of \(f(x)\) and \(g(x)\), it is clear that the asymptote of \(f(x)\) is \(y = 0\) and the asymptote of \(g(x)\) is \(y = 1\). These are horizontal lines that the graphs of the functions approach but never cross.
04

Confirm the graphs

Use a graphing utility to confirm the hand-drawn graphs. The graphing utility will show the same features identified by hand: the graphs of \(f(x)\) and \(g(x)\) decreasing as \(x\) increases, with horizontal asymptotes at \(y = 0\) and \(y = 1\) respectively.

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

$$\text { Solve for } y: 7 x+3 y=18$$

In Exercises \(141-144,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\text { If } \log (x+3)=2, \text { then } e^{2}=x+3$$

Evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 32\) b. Evaluate: \(\log _{2} 8+\log _{2} 4\) c. What can you conclude about \(\log _{2} 32,\) or \(\log _{2}(8 \cdot 4) ?\)

You take up weightlifting and record the maximum number of pounds you can lift at the end of each week. You start off with rapid growth in terms of the weight you can lift from week to week, but then the growth begins to level off. Describe how to obtain a function that models the number of pounds you can lift at the end of each week. How can you use this function to predict what might happen if you continue the sport?

Use a calculator with an \(\left[e^{x}\right]\) key to solve. The bar graph shows the percentage of U.S. high school seniors who applied to more than three colleges for selected years from 1980 through 2013. (BAR GRAPH CAN'T COPY) The data can be modeled by $$ f(x)=x+31 \text { and } g(x)=32.7 e^{0.0217 x} $$ in which \(f(x)\) and \(g(x)\) represent the percentage of high school seniors who applied to more than three colleges \(x\) years after 1980\. Use these functions to solve . Where necessary, round answers to the nearest percent. In college, we study large volumes of information \(-\) information that, unfortunately, we do not often retain for very long. The function $$ f(x)=80 e^{-0.5 x}+20 $$ describes the percentage of information, \(f(x),\) that a particular person remembers \(x\) weeks after learning the information. a. Substitute 0 for \(x\) and, without using a calculator, find the percentage of information remembered at the moment it is first learned. b. Substitute 1 for \(x\) and find the percentage of information that is remembered after 1 week. c. Find the percentage of information that is remembered after 4 weeks. d. Find the percentage of information that is remembered after one year (52 weeks).
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.