Dougherty High Graduation Test Preparation-2

1.

Inequalities and Absolute Value

Concepts "Piecewise Functions"

and Absolute Value

Kenny M. Felder

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Abstract

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. /

positive

OK, so, what happens if you put a

number back, but made positive. OK, how do you

5/ = 5. j j= . And so on. We can say, as a generalization, that j x j= x; but only if x is.negative number into an absolute value? Answer: you get that samemake a negative number positive? Mathematically, you

multiply it by 1

. j 􀀀5 j= 􀀀(􀀀5) = 5. j 􀀀 j= 􀀀(􀀀 ) = . We can say, as a generalization, that

j

So the absolute value function can be de ned like this.

x j= 􀀀x; but only ifx is negative.

The Piecewise De nition of Absolute Value

j

x j= f

x; x

0

􀀀

x; x < 0

(1)

If you've never seen this before, it looks extremely odd. If you try to pin that feeling down, I think you'll

nd this looks odd for some combination of these three reasons.

1. The whole idea of a piecewise function that is, a function which is de ned di erently on di erent

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

are more common than you might think.

2. The

But if

think of it as change the sign of

3. Even if you get past those objections, you may feel that we have taken a perfectly ordinary, easy to

understand function, and rede ned it in a terribly complicated way. Why bother?

do that? moments. Yes, it can and, in fact, functions de ned in this piecewise manner 􀀀x looks suspicious. I thought an absolute value could never be negative! Well, that's right.x is negative, then 􀀀x is positive. Instead of thinking of the 􀀀x as negative x it may help tox.

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Surprisingly, the piecewise de nition makes many problems

You already know how to graph

de nition. On the right side of the graph (where

graph (where

easier. Let's consider a few graphing problems.y =j x j. But you can explain the V shape very easily with the piecewisex 0), it is the graph of y = x. On the left side of thex < 0), it is the graph of y = 􀀀x.

(a) (b)

(c)

Figure 1:

where

where

(a) y = 􀀀 x The whole graph is shown, but the only part we care about is on the left,x < 0 (b) y = x The whole graph is shown, but the only part we care about is on the right,x 0 (c) y = j x j Created by putting together the relevant parts of the other two graphs.

Still, that's just a new way of graphing something that we already knew how to graph, right? But now

consider this problem: graph

becomes a snap.

y = x+ j x j. How do we approach that? With the piecewise de nition, it

x

+ j x j = f

x

+ x = 2x x 0

x

+ ( 􀀀 x ) = 0 x < 0

(2)

So we graph

drawings, as I did above.)

y = 2x on the right, and y = 0 on the left. (You may want to try doing this in three separate

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Figure 2:

y = x + j x j

Our nal example requires us to use the piecewise de nition of the absolute value for both

x and y.

Example 1: Graph |x|+|y|=4

We saw that in order to graph

j x j we had to view the left and right sides separately. Similarly,

j

y j divides the graph vertically.

On top, where y 0, j y j= y.

Since this equation has

horizontally and vertically, which means we look at

4

Where y < 0, on the bottom, j y j= 􀀀y.both variables under absolute values, we have to divide the graph botheach quadrant separately. j x j + j y j=

Second Quadrant First Quadrant

x

y

0, so j x j= 􀀀x x 0, so j x j= x 0, so j y j= y y 0, so j y j= y

(

􀀀x) + y = 4 x + y = 4

y

= x + 4 y = 􀀀x + 4

Third Quadrant Fourth Quadrant

x

y

0, so j x j= 􀀀x x 0, so j x j= x 0, so j y j= 􀀀y y 0, so j y j= 􀀀y

(

􀀀x) + (􀀀y) = 4 x + (􀀀y) = 4

y

= 􀀀x 􀀀 4 y = x 􀀀 4

Table 1

Now we graph each line, but only in its respective quadrant. For instance, in the fourth quadrant,

we are graphing the line

fourth quadrant.

y = x 􀀀 4. So we draw the line, but use only the part of it that is in the

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Figure 3

Repeating this process in all four quadrants, we arrive at the proper graph.

Figure 4:

j x j + j y j= 4

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