# Piecewise-Defined Function

• Jan 19th 2007, 01:05 PM
symmetry
Piecewise-Defined Function
For both questions below:

(a) Find the domain of the function.

(b) Locate any intercepts.

(1)

.....{3 + x......if -3 <or= to x < 0
f(x){3...........if......x = 0
.....{Sqrt{x}..if......x > 0

=======================

(2)

..........{1/x.........if.....x < 0
f(x) = {sqrt{x}...if.....x >or= to 0

NOTE: It is hard to correctly type the piecewise-defined functions using a regular keyboard.

I hope you can understand the above.
• Jan 19th 2007, 01:48 PM
AfterShock
Quote:

Originally Posted by symmetry
For both questions below:

(a) Find the domain of the function.

(b) Locate any intercepts.

(1)

.....{3 + x......if -3 <or= to x < 0
f(x){3...........if......x = 0
.....{Sqrt{x}..if......x > 0

=======================

(2)

..........{1/x.........if.....x < 0
f(x) = {sqrt{x}...if.....x >or= to 0

NOTE: It is hard to correctly type the piecewise-defined functions using a regular keyboard.

I hope you can understand the above.

I'll do the first one for you- graph it. The conditions are the "if" parts in the piece-wise function. Domain is (-3, inf) and there are no intersepts. Try graphing it. You have a line with slope = 1 and an exponential function.

EDIT: Sorry, there are x and y-intercepts, as Soroban pointed out, although the two graphs do not not intersect which is what I was getting at.
• Jan 19th 2007, 02:16 PM
Soroban
Hello, symmetry!

Quote:

For both questions below:
. . (a) Find the domain of the function.
. . (b) Locate any intercepts.

Did you make a sketch?

Quote:

$\displaystyle (1)\;\;f(x) \:=\:\begin{Bmatrix} 3 + x & &\text{if }\text{-}3 \leq x < 0 \\ 3 & &\text{if }x = 0 \\ \sqrt{x} & &\text{if }x > 0 \end{Bmatrix}$
Code:

                |                 |            *                 *      *               * |  *             *  |*       ----*-----o--------------         -3    |
Domain: .$\displaystyle (\text{-}3,\,\infty)$

Intercepts: .$\displaystyle (\text{-}3,0),\:(0,3)$

Quote:

$\displaystyle (2)\;\;f(x)\:=\:\begin{Bmatrix}\frac{1}{x} & & \text{if }x < 0 \\ \sqrt{x} & & \text{if }x \geq 0 \end{Bmatrix}$
Code:

                      |                       |            *                       |      *                       |  *                       |*     ------------------*----------------       *              |             *        |                 *    |                   *  |                     * |                       |                     *|                       |
Domain: .$\displaystyle (-\infty,\,\infty)$

Intercepts: .$\displaystyle (0,\,0)$

• Jan 19th 2007, 06:42 PM
symmetry
ok
Thank you again both for your quick replies.

To soroban,

No, I did not sketch the graph because I do not know how to graph piecewise-defined functions.

I understand these functions are graphed in parts, right?

Can you take me through a sample graphing question in terms of this type of function?

Thanks!
• Jan 19th 2007, 11:49 PM
AfterShock
Quote:

Originally Posted by symmetry
Thank you again both for your quick replies.

To soroban,

No, I did not sketch the graph because I do not know how to graph piecewise-defined functions.

I understand these functions are graphed in parts, right?

Can you take me through a sample graphing question in terms of this type of function?

Thanks!

Yes, they are 'graphed in parts,' I guess you could call it.

For instance,

Take the first condition;

f(x) = 3 + x if -3 <= x < 0

From x = -3 (including this point) to x = 0 (not including, and thus draw an open circle by this point), you will graph 3 + x; see Soroban's graph. The reason why it's closed (solid dot) is because of the next condition later, and thus includes that point. Try look up piece-wise functions on Wikipedia.
• Jan 20th 2007, 05:34 AM
symmetry
ok
I like graphing functions. I think piecewise-defined functions are cool but not easy to sketch.

Thanks!
• Jan 20th 2007, 06:05 AM
Soroban
Hello again, symmetry!

Okay, here's an example.

. . $\displaystyle f(x) \:=\:\begin{Bmatrix}3 & \text{if }0 \leq x \leq 1 \\ 2x + 1 & \text{if }x > 1\end{Bmatrix}$

When $\displaystyle x$ is between $\displaystyle 0$ and $\displaystyle 1$ (including the endpoints),
. . the graph is $\displaystyle f(x) = 3$, a horizontal line.
Code:

        |       3* * * * *         |         |       - + - - - + - -         |      1

When $\displaystyle x$ is greater than 1, the graph is $\displaystyle f(x) \:=\:2x + 1$,
. . a slanted line.
Code:

        |         |                  *         |                *         |            *         |          *       3|      *         |         |       - + - - - + - - - - - - -         |      1

Sketch them on the same graph
. . and have the graph of the piecewise function.
Code:

        |         |                  *         |                *         |            *         |          *       3o * * * *         |         |       - + - - - + - - - - - - -         |      1

This function could be your long-distance charge.

They might charge $3 for the first minute . . and$2 per minute for every subsequent minute.

(Hmmm, not a good example . . .
I'm sure someone will point out why.)