Q. 69 - Precalculus Enhanced with Graphing Utilities

Question

Define some functions that pass through $(0,0)$ and $(1,1)$ and are increasing for $x\ge 0$. Begin your list with $y=\sqrt{x},y=x$, and $y=x^2$. Can you propose a general result about such functions? 

Step-by-Step Solution

Answer

All functions of the form $y=x^k$, $k>0$ passes through the points $(0,0)$ and $(1,1)$ and are increasing for $x\ge 0$. As the exponent $k$ increases the steepness of the curve increases.

Step 1. Define the functions

The general function rule for the functions passing through the points $(0,0)$ and $(1,1)$ and are increasing for $x\ge 0$ is given as $y=x^k$ where $k>0$.

Some such functions are 

$$\begin{gathered} y=\sqrt{x} \\ y=x \\ y=x^2 \\ y=x^3 \end{gathered}$$

and so on.

Step 2. General result

For the function $y=x^k$, $k>0$ as the exponent $k$ increases the steepness of the curve also increases.