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.
.
. And so on. We can say, as a generalization, that
; but
only ifis 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 .
.
. We can say, as a generalization, that
; but
only ifis negative .
So the absolute value function can be defined like this.
The “piecewise” definition of absolute value
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.
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.
The
looks suspicious. “I thought an absolute value could never
be negative!” Well, that’s right. But if
is negative, then
is positive. Instead of thinking of the
as “negative
” it may help to think of it as “change the sign of
.”
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
. But you can explain the V shape very easily with the piecewise definition. On the right side of the graph (where
), it is the graph of
. On the
left side of the graph (where
), it is the graph of
.
The whole graph is shown, but the only part we care about is on the left, where
The whole graph is shown, but the only part we care about is on the right, where
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
. How do we approach that? With the piecewise definition, it becomes a snap.
So we graph
on the right, and
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
and
.
Graph |x|+|y|=4
We saw that in order to graph
we had to view the left and right sides separately. Similarly,
divides the graph
vertically .
On top, where
,
.
Where
, on the bottom,
.
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 .
Second Quadrant
First Quadrant
, so
, so
, so
, so
Third Quadrant
Fourth Quadrant
, so
, so
, so
, so
Now we graph each line, but only in its respective quadrant. For instance, in the fourth quadrant, we are graphing the line
. 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.
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .