Chapter 2.4, Problem 69AYU

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

To determine

To graph: Graph the equations $y = \sqrt{x}, y = x,$ and $y = x^2$. Then conclude common things about each graph.

Answer to Problem 69AYU

Explanation of Solution

To sketch:

Graph the following equation with the graphing utility:

1. $y = \sqrt{x}$

2. $y = x$

3. $y = x^2$

Interpretation:

All the three equations $y = \sqrt{x}, y = x,$ and $y = x^2$. are powers of $x$ because $1^{any thing} = 1$.

The equations $y = \sqrt[3]{x}, and y = x^3$ are also behaves like the other functions.

In general, $y = x^n$, n is any real number behaves like such functions.