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Identity function

The dependent (y) and independent (x) variables have same value. Identity function is similar in concept to that of identity relation which consists of relation of an element of a set with itself. It is a linear function in which m=1 and c=0. Identity function form is represented as :

y = f(x) = x

Identity function

Identity function is a polynomial of degree 1.

The graph of identity function is a straight line bisecting first and third quadrants of coordinate system. Note that slope of straight line is 45°. It is clear from the graph that its domain and range both are real number set R.

Quadratic function

The general form of quadratic function is :

f(x) = a x 2 + b x + c ; a , b , c R ; a 0

We shall discuss quadratic function in detail in a separate module and hence discussion of this function is not taken up here.

Graph of polynomial function

Graph of polynomial is continuous and non-periodic. If degree is greater than 1, then it is a non-linear graph. Polynomial graphs are analyzed with the help of function properties like intercepts, slopes, concavity, and end behaviors. The may or may not intersect x-axis. This means that it may or may not have real roots. As maximum number of roots of a polynomial is at the most equal to the order of polynomial, we can deduce that graph can at the most intersect x-axis “n” times as maximum numbers of real roots are “n”.

The fact that graph of polynomial is continuous suggests two interesting inferences :

1: If there are two values of polynomial f(a) and f(b) such that f(a)f(b)<0, then there are at least 1 or an odd numbers of real roots between a and b. The condition f(a)f(b)<0 means that function values f(a) and f(b) lie on the opposite sides of x-axis. Since graph is continuous, it is bound to cross x-axis at least once or odd times. As such, there are at least 1 or odd numbers of real roots (as shown in the left figure down).

Roots of polynomial function

The numbers of x-intercepts depend on nature of product given by f(a)f(b).

2 : If there are two values of polynomial f(a) and f(b) such that f(a)f(b)>0, then there are either no real roots or there are even numbers of real roots between a and b. The condition f(a)f(b)>0 means that function values f(a) and f(b) are either both negative or both positive i.e. they lie on the same side of x - axis. Since graph is continuous, it may not cross at all or may cross x-axis even times (as shown in the right figure above). Clearly, there is either no real root or there are even numbers of real roots.

We shall study graphs of quadratic polynomials in a separate module. Further, other graphs will be discussed in appropriate context, while discussing a particular function. Here, we present two monomial quadratic graphs y = x 2 and y = x 3 . These graphs are important from the point of view of generalizing graphs of these particular polynomial structure. The nature of graphs y = x n , where “n” is even integer greater than equal to 2, is similar to the graph of y = x 2 . We should emphasize that the shape of curve simply generalizes the nature of graph – we need to draw them actually, if we want to draw graph of a particular monomial function. However, we shall find that these generalizations about nature of curve lets us know a great deal about the monomial polynomial. In particular, we can conclude that their domain and range are real number set R.

Even degree function

The nature of graphs of degree of positive even integer are similar to the graph shown.

Similarly, the nature of graphs y = x n , where “n” is odd number integer greater than 2, is similar to the graph of y = x 3 .

Odd degree function

The nature of graphs of degree of positive odd integer are similar to the graph of shown.

Questions & Answers

what are two classical macroeconomics and what're their theories say about their equations?
what is the formula for calculating elasticity
aza Reply
what is elasticity of demand?
Rita Reply
Causes of economic growth
pierre Reply
What is elasticity of demand
What are the causes of economic growth
economic growth, establishment of industry, encourage of investor's, farm productivities, creation of institutions, construction of good road etc
elasticity of demand can be said to be the responsiveness of demand to a change in prices
impact of collusion in the economy referring to inefficiencies illustrated by means of graph
nondumiso Reply
The Factor price will determine the choice of techniques to produce.Expantiate
what is elasticity of demand?
Etta Reply
state and explain two types of demand
Institution involved in money market
Gande Reply
what is Economics
Kwame Reply
Economic is the study of scarcity
Economics is the study of a lot of things. It is split up into two areas of study, Microeconomics and Macroeconomics. Microeconomics is the study of an individual's choices in the economy and Macroeconomics is the study of the economy as a whole.
Economics is a science that studies human scarcity
What is Equilibrium price?
Equilibrium is the market clearing price. The point at which quantity demanded equals quantity supplied. The point at which the supply and demand curves intersect.
Equilibrium price*
Refers to the study of how producers use limited resources to satisfy human unlimited wants
why is economics important
Derrick Reply
What will you do as a consumer if you are not at equilibrium?
chukwu Reply
am new I will like to know about the graph relationship
Gloria Reply
comment on WTO principle on trading system. trade without discrimination
Omben Reply
optimize z=f(x,y)=6x²-9x-3xy-7y+5y²
Alex Reply
What is an indifference curve?
layla Reply
different levels of utilities of a person in a given set of bundles of goods
identify and quantify five social costs and social benefits of building a school
Mokgobo Reply
identify and quantity five social costs and social benefits of building a hospital
short run vs long run
state the law of diminishing return?
The Law of Diminishing (Marginal) Returns simply states that at some point in time a business/operation/etc.'s increased productivity will begin to decline.
For example, if a small pizza shop currently has 3 workers in the kitchen at any given time,and hiring 1 more worker will increase productivity, at some number of workers hired will the business see a decrease in productivity because the capital resources that the pizza shop has is not infinite.
Five social benefits of building a hospital, in my opinion and depending on where it's built, would be 1) Increased care for neighboring residents, 2) Potential jobs for individuals, 3) May decrease the travel time residents need to endure in order to reach the nearest hospital
4) May create work-study programs for individuals who aspire to be future Doctors, Nurses, Physicians, etc. 5) Assuming there are local pharmaceutical businesses nearby, the hospital may decide to purchase supplies local, increasing the business' sales. Thus, generating more income.
5 costs of building a hospital would be 1) Increased noise and waste pollution from service vehicles and hospital visitors, 2) May require large amounts of space, possibly jeopardizing nearby animal habitats, 3) May see an increase in traffic and possibly car accidents from frantic individuals
racing to see their injured friends, family members, etc. 4) Constructing a hospital and hiring staff is very expensive 5) To use funds, private or public, to finance the construction of a hospital cannot be used to fund any other projects. (The concept of opportunity costs.)
what is meant by inteference with the price mechanism operation?
We use a Supply and Demand graph to illustrate at what price level will the market for a certain good or service be at equilibrium. If the price for a good or service is set too high, consumers will be less inclined to buy that product Thus, creating a surplus.
This surplus will eventually drive the price back down to it's equilibrium point. Similarly, if a price for a good or service is set too low, individuals would be more inclined to buy more of a certain product, creating a shortage. This shortage will cause sellers to drive the price back up to the
equilibrium point.
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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