# 2.5 "piecewise functions" and absolute value

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This module introduces piecewise functions for the purpose of understanding absolute value equations.

What do you get if you put a positive number into an absolute value? Answer: you get that same number back. $\mid 5\mid =5$ . $\mid \pi \mid =\pi$ . And so on. We can say, as a generalization, that $\mid x\mid =x$ ; but only if $x$ is positive .

OK, so, what happens if you put a negative number into an absolute value? Answer: you get that same number back, but made positive. OK, how do you make a negative number positive? Mathematically, you multiply it by –1 . $\mid -5\mid =-\left(-5\right)=5$ . $\mid -\pi \mid =-\left(-\pi \right)=\pi$ . We can say, as a generalization, that $\mid x\mid =-x$ ; but only if $x$ is negative .

So the absolute value function can be defined like this.

## The “piecewise” definition of absolute value

$\mid x\mid =\left\{\begin{array}{cc}x,& x\ge 0\\ -x,& x<0\end{array}$

If you’ve never seen this before, it looks extremely odd. If you try to pin that feeling down, I think you’ll find this looks odd for some combination of these three reasons.

1. The whole idea of a “piecewise function”—that is, a function which is defined differently on different domains—may be unfamiliar. Think about it in terms of the function game. Imagine getting a card that says “If you are given a positive number or 0, respond with the same number you were given. If you are given a negative number, multiply it by –1 and give that back.” This is one of those “can a function do that ?” moments. Yes, it can—and, in fact, functions defined in this “piecewise manner” are more common than you might think.
2. The $-x$ looks suspicious. “I thought an absolute value could never be negative!” Well, that’s right. But if $x$ is negative, then $-x$ is positive. Instead of thinking of the $-x$ as “negative $x$ ” it may help to think of it as “change the sign of $x$ .”
3. Even if you get past those objections, you may feel that we have taken a perfectly ordinary, easy to understand function, and redefined it in a terribly complicated way. Why bother?

Surprisingly, the piecewise definition makes many problems easier . Let’s consider a few graphing problems.

You already know how to graph $y=\mid x\mid$ . But you can explain the V shape very easily with the piecewise definition. On the right side of the graph (where $x\ge 0$ ), it is the graph of $y=x$ . On the left side of the graph (where $x<0$ ), it is the graph of $y=-x$ .

Still, that’s just a new way of graphing something that we already knew how to graph, right? But now consider this problem: graph $y=x+\mid x\mid$ . How do we approach that? With the piecewise definition, it becomes a snap.

$x+|x|=\left\{\begin{array}{cc}x+x=2x\hfill & x\ge 0\\ x+\left(-x\right)=0& x<0\end{array}$

So we graph $y=2x$ on the right, and $y=0$ on the left. (You may want to try doing this in three separate drawings, as I did above.)

Our final example requires us to use the piecewise definition of the absolute value for both $x$ and $y$ .

## Graph |x|+|y|=4

We saw that in order to graph $\mid x\mid$ we had to view the left and right sides separately. Similarly, $\mid y\mid$ divides the graph vertically .

• On top, where $y\ge 0$ , $\mid y\mid =y$ .
• Where $y<0$ , on the bottom, $\mid y\mid =-y$ .
Since this equation has both variables under absolute values, we have to divide the graph both horizontally and vertically, which means we look at each quadrant separately . $\mid x\mid +\mid y\mid =4$
 Second Quadrant First Quadrant $x\le 0$ , so $\mid x\mid =-x$ $x\ge 0$ , so $\mid x\mid =x$ $y\ge 0$ , so $\mid y\mid =y$ $y\ge 0$ , so $\mid y\mid =y$ $\left(-x\right)+y=4$ $x+y=4$ $y=x+4$ $y=-x+4$ Third Quadrant Fourth Quadrant $x\le 0$ , so $\mid x\mid =-x$ $x\ge 0$ , so $\mid x\mid =x$ $y\le 0$ , so $\mid y\mid =-y$ $y\le 0$ , so $\mid y\mid =-y$ $\left(-x\right)+\left(-y\right)=4$ $x+\left(-y\right)=4$ $y=-x-4$ $y=x-4$
Now we graph each line, but only in its respective quadrant. For instance, in the fourth quadrant, we are graphing the line $y=x-4$ . So we draw the line, but use only the part of it that is in the fourth quadrant. Repeating this process in all four quadrants, we arrive at the proper graph.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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?
Kyle
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what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
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characteristics of micro business
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there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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