Problem 49 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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).

Step 2: 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 \(f\).

Step 3: 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).