# Graph Transformation Problems

• Jan 16th 2007, 04:55 PM
aussiekid90
Graph Transformation Problems
1. If a curve on a graph of f is F9x)=x^2-2x for all real numbers x, is the absolute value (f(x)) diffentiable at x=0? I'm pretty sure its no, but i'm not sure how to explain it.

2. If the same curve exists, is the f(absolute value(x)) differentiable at x=0? Once again, i believe its no, but i can't explain it well.

Thanks a lot

-aussiekid90
• Jan 16th 2007, 05:08 PM
ThePerfectHacker
Quote:

Originally Posted by aussiekid90
1. If a curve on a graph of f is F9x)=x^2-2x for all real numbers x, is the absolute value (f(x)) diffentiable at x=0? I'm pretty sure its no, but i'm not sure how to explain it.

2. If the same curve exists, is the f(absolute value(x)) differentiable at x=0? Once again, i believe its no, but i can't explain it well.

Consider the general absolute value $y=|x|$.
Is it differenciable at $x=0$?
No.

Because, at $x=0$ we have.
$\lim_{\Delta x\to 0}\frac{|\Delta x|}{\Delta x}$
If we approach from the right we have 1.
If we approach from the left we have -1.
The sided limits do not conincide.
• Jan 16th 2007, 07:01 PM
Soroban
G'day, aussiekid90!

Quote:

1. If $f(x)\:=\:x^2-2x$, is $|f(x)|$ differentiable at $x=0$?
The graph of $f(x) \:=\:x^2-2x$ looks like this:
Code:

            * |              *               |             *|              *       --------*-------------*--             0| *        * 2               |    * *               |
In the graph of $g(x) \:=\:|x^2-2x|$,
. . anything below the x-axis is reflected upward.
Code:

            * |              *               |    * *             *| *      *  *       --------*-----------*--               |
The graph has two different slopes at $x = 0$.
. . The function is not differentiable there.

Quote:

2. For the same curve, is $f(|x|)$ differentiable at $x=0$?

The graph of $f(|x|)$ is symmetric to the y-axis.
The graph looks like this:
Code:

      *              |              *                       |       *              |              *       --*-------------*-------------*--           *        * | *        *               * *    |    * *                       |

The graph has two different slopes at $x=0$.
. . The function is not differentiable there.