Problem 57 - College Algebra Essentials

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 point of intersection of the two graphs is \(x=0\), \(y=1\).

Step 1: Graph the function f(x)

The function \(f(x)=2^x\) is a classic example of an exponential function. It increases as x increases, starting from \(y=1\) when \(x=0\). It will be traced on the positive y-axis.

Step 2: Graph the function g(x)

The function \(g(x)=2^{-x}\) is an exponential function mirrored along the y-axis compared to \(f(x)\). It decreases as x increases, starting from \(y=1\) when \(x=0\). It will mostly be traced on the positive y-axis.

Step 3: Find the point of intersection

Look for the point where the two graphs cross each other. One graph increases and the other decreases as x increases. Since both start from the same point when \(x=0\), they intersect at that point. So, the intersection is at \(x=0\), \(y=1\). This can be also confirmed by setting \(f(x)=g(x)\) and solving for \(x\). When \(x=0\), \(f(0)=g(0)=1\).