Problem 51 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph \(y=2^{x}\) and \(x=2^{y}\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

After graphing the two functions, \(y=2^{x}\) appears as a curve passing through (0,1) that increases as x increases. The graph of \(x=2^{y}\) (or \(y=\log_{2}{x}\)) starts from the y-axis and increases as y increases. Both graphs intersect at two points: (0,1) and (1,2).

Step 1 - Graph the function \(y=2^{x}\)

To graph the function \(y=2^{x}\), we start by creating a table of values then plot these values on the rectangular coordinate system. The exponential function \(y=2^{x}\) will result in a graph that is always above the x-axis, passes through the point (0,1) and increases as x increases.

Step 2 - Graph the function \(x=2^{y}\)

To graph the function \(x=2^{y}\), we first rewrite it as \(y=\log_{2}{x}\), by applying the logarithm definition. Then similar to the previous step, we create a table of values and then plot these values in the same rectangular coordinate system. The resulting graph starts from the y-axis and increases as y increases.

Step 3 - Identify Intersections

By comparing the two functions visually on the graph, we can see where they intersect. These intersections provide the solutions to the equation \(2^{x} = \log_{2}{x}\).