Problem 92 - Algebra and Trigonometry

Question

Graph \(f(x)=2^{x}\) and its inverse function in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The graphs of \(f(x) = 2^{x}\) and its inverse function \(f^{-1}(x) = log_{2}{x}\) should be plotted on the same coordinate system. The former is an increasing curve crossing the y-axis at y=1. The latter is also an increasing curve, but crossing the x-axis at x=1. One can clearly see that both graphs are mirror images of each other about the line y=x.

Step 1: Understanding the function

The function given is \(f(x) = 2^{x}\), which is an exponential function. For any real number x, 2 to the power of x is defined.

Step 2: Plot the function

Create a table of values to help determine points on the graph of \(f(x) = 2^{x}\). Then use these points to sketch the graph. Include at least a few negative and positive values of x.

Step 3: Finding the inverse function

The inverse function of \(f(x) = 2^{x}\) is found by swapping the roles of y and x. We write \(y = 2^{x}\) as \(x = 2^{y}\), which gives \(y = log_{2}{x}\). So, the inverse function is \(f^{-1}(x) = log_{2}{x}\).

Step 4: Plot the inverse function

Similarly as in Step 2, create a table of values to help determine points on the graph of \(f^{-1}(x) = log_{2}{x}\). Then use these points to sketch the graph.

Step 5: Include both graphs

Graph \(f(x) = 2^{x}\) and \(f^{-1}(x) = log_{2}{x}\) in the same rectangular coordinate system.