Problem 103 - College Algebra

Question

Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. $$y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2}$$

Step-by-Step Solution

Answer

The functions in order from the slowest to the most rapidly increasing are as follows: \(y=\ln x\), \(y=\sqrt{x}\), \(y=x\), \(y=x^{2}\), \(y=e^{x}\), \(y=x^{x}\).

Step 1: Graphing each function

Plot the functions in the same viewing area. An understanding on how the functions look graphically is important, so start by plotting: \(y=x\), \(y=\sqrt{x}\), \(y=e^{x}\), \(y=\ln x\), \(y=x^{x}\), and \(y=x^{2}\). Tools like a calculator or suitable software can be used for plotting.

Step 2: Studying the behavior of each function

With the graphs plotted, observe their behavior as x increases. Since these functions don't have the same values of x and y, the rates at which y-values change as x-values increase will have to allow to rank the functions. Here are some general observations: \(y=x\) is a straight line, which increases at a constant rate. \(y=\sqrt{x}\) increases slower and always less than x. \(y=e^{x}\) increases quite quickly and always more than x. \(y=\ln x\) increases very slowly, even slower than sqrt(x). \(y=x^{x}\) increases rapidly, even more than e^x for large x, and \(y=x^{2}\) increases more rapidly than x and sqrt(x) but less rapidly than e^x and x^x.

Step 3: Arranging the functions in order

After an examination of the behaviors of each function, you can place them in order from the slowest to the most rapidly increasing. Use the observations made in the previous step to do so. The order is as follows: \(y=\ln x\), \(y=\sqrt{x}\), \(y=x\), \(y=x^{2}\), \(y=e^{x}\), \(y=x^{x}\).