Problem 108 - Introductory and Intermediate Algebra for College Students 4th

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 slowest to fastest increase is: \(y=\sqrt{x}\), \(y=\ln x\), \(y=x\), \(y=x^{2}\), \(y=e^{x}\), \(y=x^{x}\).

Step 1: Sketch the Graphs

Plotting can be done either manually or by use of a graphing calculator. Here are the steps to sketching each function:\n 1. \(y=x\): This is a simple straight line passing through the origin.\n 2. \(y=\sqrt{x}\): The graph of the square root of x is a curve starting from the origin, which increases more slowly than \(y=x\).\n 3. \(y=e^{x}\): The graph is an exponential curve which goes through (0,1). It increases faster as x increases. \n 4. \(y=\ln x\): The natural logarithm graph slowly increases and it is undefined for \(x\leq 0\).\n 5. \(y=x^{x}\): This function increases very slowly for \(x<1\), and then starts to increase more rapidly. The function is undefined for \(x\leq 0\).\n 6. \(y=x^{2}\): A parabola opening upwards with its vertex at the origin. It increases more rapidly than \(y=x\) but slower than \(y=e^{x}\).

Step 2: Rank the Functions

Based on the graphs and behavior of each function, we can rank them from slowest to fastest increase as follows:\n 1. \(y=\sqrt{x}\)\n 2. \(y=\ln x\)\n 3. \(y=x\)\n 4. \(y=x^{2}\)\n 5. \(y=e^{x}\)\n 6. \(y=x^{x}\)