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Chapter 3: Problem 57

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

Step by step solution

01

Graph the functions

To draw the graphs, select a few points for the x value and calculate the corresponding y values for each function. Then plot these points and connect them to form the continuous graph for each function.
02

Redefine the function g(x)

Notice that \(g(x)=2^{-x}\) can be rewritten as \(\frac{1}{2^{x}}\). This is because the negative exponent rule states that \(a^{-n}=\frac{1}{a^n}\), where \(a\) is the base and \(n\) is the exponent.
03

Setting f(x) equal to g(x)

Set \(2^{x}=\frac{1}{2^{x}}\). Since neither side is zero, we can safely cross-multiply to clear the fraction and get \(2^{2x}=1\).
04

Solve the equation

To solve for x, observe that \(2^{2x}\) will be equal to 1 only when \(2x = 0\), implying that \(x=0\). So the point of intersection is at \(x=0\).
05

Find the corresponding y values

Substitute \(x=0\) into either of the original equations to find the corresponding y value. Thus, \(y = 2^{0}\) or \(y = 2^{-0}\), both of which are equal to 1. So the point of intersection is at (0, 1).

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Most popular questions from this chapter

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