Problem 134 - 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 order from the function that increases most slowly to the one that increases most rapidly is: \(y = \ln x\), \(y = \sqrt{x}\), \(y = x\), \(y = x^{2}\), \(y = e^{x}\), \(y = x^{x}\).

Step 1: Graph the functions

Begin by graphing the first function, \(y = x\). This is a simple linear function and its graph is a straight line with a slope of one. Then, graph the second function, \(y = \sqrt{x}\). This function is slower growing than \(y = x\). Continue this process for the remaining functions: \(y = e^{x}\), \(y = \ln x\), \(y = x^{x}\), \(y = x^{2}\). These graphs will form the basis for inspecting and comparing the growth rates of the functions.

Step 2: Analyze the graphs

Once all of the functions have been graphed, the next step is to analyze the graphs in order to rank the functions. Start by identifying the function that increases most slowly. Then, sequentially identify functions that increase faster than the previous one.

Step 3: Order the functions

After careful analysis, it can be concluded that the order from the one that increases most slowly to the one that increases most rapidly is: \(y = \ln x\), \(y = \sqrt{x}\), \(y = x\), \(y = x^{2}\), \(y = e^{x}\), \(y = x^{x}\).