/*! 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 49 Graph \(f\) and \(g\) in the sam... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 49

Graph \(f\) and \(g\) in the same rectangular coordinate system. Then find the point of intersection of the two graphs. $$f(x)=2^{x}, g(x)=2^{-x}$$

Short Answer

Expert verified
The functions \(f(x)=2^{x}\) and \(g(x)=2^{-x}\) intersect at the point (0,1).

Step by step solution

01

Graph the function \(f(x)=2^{x}\)

First, graph the function \(f(x)=2^{x}\). This is a simple exponential function with a base of 2. Generally, all exponential functions with a base greater than 1 grow rapidly as x increases and get very close to 0 as x decreases. The graph of \(f(x)=2^{x}\) cuts the y-axis at the point (0,1).
02

Graph the function \(g(x)=2^{-x}\)

Next, graph the function \(g(x)=2^{-x}\). This function can also be written as \(g(x)=\frac{1}{2^x}\), which shows that it is the inverse function of \(f\). The graph of \(g\) will be a reflection of the graph of \(f\) about the line y=x. Hence it cuts the y-axis at the same point (0,1) as \(f\).
03

Find the point of intersection

The point of intersection of \(f\) and \(g\) can be found by setting \(f(x) = g(x)\) and solving for \(x\). This gives \(2^x = 2^{-x}\), which simplifies to \(2^x = \frac{1}{2^x}\) or \(2^{2x} = 1\). The only value of \(x\) for which this equation is true is \(x = 0\). Thus the point of intersection is at the point where x is 0 and y is \(f(0) = g(0) = 1\), i.e., the point (0,1).

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

Solve: $$\sqrt{2 x-1}-\sqrt{x-1}=1$$

$$\text { Solve: } x-2(3 x-2)>2 x-3$$

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

Describe the following property using words: \(\log _{b} b^{x}=x\)

Complete the table for a savings account subject to continuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. $$\begin{array}{l|c|l|c} \hline \begin{array}{l} \text { Amount } \\ \text { Invested } \end{array} & \begin{array}{l} \text { Annual Interest } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Accumulated } \\ \text { Amount } \end{array} & \begin{array}{l} \text { Time } t \\ \text { in Years } \end{array} \\ \hline \$ 2350 & & \text { Triple the amount invested } & 7 \\ \hline \end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied 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.