Problem 59 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

After graphing the functions and finding the intersection, the intersection point was found to be (2,2). That is where the plots of \(y=2^{x}\) and \(x=2^{y}\) cross each other.

Step 1: Graphing the first function

The equation is \(y=2^{x}\). This is a basic exponential function. The graph starts from the point (0,1), since any number raised to 0 is 1. As x gets larger, y also gets larger. Start plotting this graph in the coordinate system.

Step 2: Graphing the second function

The equation is \(x=2^{y}\). This is not a common form of a function but it still can be graphed. It is helpful to think of it as \(y=\log_{2}{x}\). The graph starts from the point (1,0), and as x gets larger, y also gets gradually larger. Plot this graph on the same coordinate system.

Step 3: Find the point(s) of intersection

The intersection points are where the graphs of \(y=2^{x}\) and \(x=2^{y}\) cross each other. Inspect the drawn graphs and determine the specific points where the two curves intercept.