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Chapter 3: Problem 58

Graph \(f\) and \(g\) in the same rectangular coordinate system. Then find the point of intersection of the two graphs. $$f(x)=2^{x+1}, g(x)=2^{-x+1}$$

Short Answer

Expert verified
After graphing both functions and finding where they intersect, it can be concluded that \(f(x)\) and \(g(x)\) intersect at \(x=0\).

Step by step solution

01

Understanding Function Properties

The given functions are of the form \( f(x)=a^x \). These are known as exponential functions. The function \(f(x) = 2^{x + 1}\) can be considered as the function \(f(x) = 2^x\) shifted one unit to the left, and the function \(g(x) = 2^{-x + 1}\) can be seen as the function \(g(x) = 2^{-x}\) shifted one unit to the right. These shifts are the result of the '+1' and '-1' in the exponents.
02

Plotting the Graphs

Plot the function \(f(x)=2^{x+1}\) by firstly calculating values for a few different \(x\) values. Similarly, calculate the \(y\) values for \(g(x)=2^{-x+1}\) using the same \(x\) values. Then, plot these points on the graph. Both graphs will appear as curves, with the graph of \(f(x)\) moving from left to right, starting low and ending high. The graph for \(g(x)\) moves in the opposite direction, i.e., from right to left starting high and ending low.
03

Finding Intersection Point

The intersection point of the two graphs is the point where they cross each other. Looking at the plotted graphs, find the point where the two curves cross. That point represents the x value for which both \(f(x)\) and \(g(x)\) have the same value.

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Most popular questions from this chapter

The exponential growth models describe the population of the indicated country, \(A\), in millions, \(t\) years after 2006 $$\begin{array{l}\mathrm{Camada}\quadA=33.1e^{0.009\mathrm{t}}\\\\\mathrm{U}_{\mathrm{ganda}}\quad A=28.2 e^{0.034 t}\end{array}$$ In Exercises \(81-84,\) use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The models indicate that in \(2013,\) Uganda's population will exceed Canada's.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph of \(f(x)=3 \cdot 2^{x}\) shows that the horizontal asymptote for \(f\) is \(x=3\)

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Exercises \(153-155\) will help you prepare for the material covered in the next section. The formula \(A=10 e^{-0.003 t}\) models the population of Hungary, \(A\), in millions, \(t\) years after 2006 a. Find Hungary's population, in millions, for 2006,2007 \(2008,\) and \(2009 .\) Round to two decimal places. b. Is Hungary's population increasing or decreasing?
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