sketching the graph of functions Good morning!

How to draw the graph of the following function and discuss the increasing and decreasing intervals of y. 

$y = $x$  + \sqrt[]{x^2}$

Also, I would like to know free online graphing calculator

Thanks in advance.
Hint: Recall that $\sqrt{x^2} = |x|$, so

$$f(x) = x + |x| = \left\{\begin{array}{lr} 2x &amp;: \text{ if } x \ge 0 \\ 0 &amp;: \text{ if } x &lt; 0 \end{array}\right.$$



For an online graphic calculator, I would suggest Wolfram Alpha.
You can try Calcpad, it is free for both online and desktop: 

http://calcpad.net/Calculator

Just type the following:

f(x) = x + sqr(x^2)

$Plot{f(x)  @ x = -1 : 1}

This is the graph you will get

Plot 1

You can also define a function with three segments as follows:

f(x) = -2*(x > 0) + 0.5*(x ≡ 0) - x^3 * (x &lt; 0)

The conditions in the brackets return 1 for "true" and 0 for "false" which makes possible to create an expression like the above one. It will evaluate correctly: 

f(-2) = 8

f(0) = 0.5

f(2) = -2

However, it will not plot correctly as one function. You have to use the "multiple" plot command. It has the following syntax:

$Plot{x1 | y1 &amp; x2 | y2 &amp; x3 | y3 @ x = a : b}

For your example, type the following...

$Plot{x*(x > 0) | -2 &amp; 0 | 0.5 &amp; x*(x &lt; 0) | -x^3*(x &lt; 0) @ x = -1 : 1}

...and you will get this plot:

Plot 2