Problem 134 - Algebra and Trigonometry

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

Order of Functions from slowest increase to fastest increase: \[y=\ln x < y=\sqrt{x} < y=x < y=x^{2}< y=e^{x} < y=x^{x}\]

Step 1: Graphing the Functions

First, graph each function in the same viewing rectangle. Using graphing software or a graphing calculator can significantly simplify this process. It's important to graph each function individually and understand its basic shape.

Step 2: Observation of Individual Graphs

Once each function is graphed, observe each individual graph. Notice how rapidly each function increases. Some functions like y=\(\ln x\) are growing slowly pretty continuously. Others like y=\(x^{x}\) increase more exponentially.

Step 3: Rank the Functions by Rate of Increase

Now, rank each of the functions based on the observation in Step 2. The function that increases the most slowly should go first and the one that increases the most rapidly should go last.