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Inequality is an important concept in understanding function and its properties – particularly domain and range. Many function forms are valid in certain interval(s) of real numbers. This means definition of function is subjected to certain restriction of values with respect to dependent and independent variables. The restriction is generally evaluated in terms of algebraic inequalities, which may involve linear, quadratic, higher degree polynomials or rational polynomials.

Function definition and inequality

A function imposes certain limitations by virtue of definition itself. We have seen such restriction with respect to radical functions in which polynomial inside square root needs to be non-negative. We have also seen that denominator of a rational function should not be zero. We shall learn about different functions in subsequent modules. Here, we consider few examples for illustration :

1 : f x = log a 3 x 2 x + 4

Here, logrithmic function is defined for a 0,1 1, and

3 x 2 x + 4 > 0

2 : f x = arcsin 3 x 2 x + 4

Here, arcsine function is defined in the domain [-1,1]. Hence,

- 1 3 x 2 x + 4 1

It is clear that we need to have clear understanding of algebraic inequalities as function definitions are defined with certain condition(s).

Forms of function inequality

Function inequality compares function to zero. There are four forms :

1 : f(x)<0

2 : f(x) ≤ 0

3 : f(x)>0

4 : f(x) ≥ 0

Here, f(x)<0 and f(x)>0 are strict inequalities as they confirm the notion of “less than” and “greater than”. There is no possibility of equality. If a strict inequality is true, then non-strict equality is also true i.e.

1 : If f(x)>0 then f(x) ≥ 0 is true.

2 : If f(x) ≥ 0 then f(x)>0 is not true.

3 : If f(x)<0 then f(x) ≤ 0 is true.

4 : If f(x) ≤ 0 then f(x)<0 is not true.

Further, we may be presented with inequality which compares function to non-zero value :

3 x 2 x - 4

However, such alterations are equivalent expressions. We can always change this to standard form which compares function with zero :

3 x 2 x + 4 > 0


Some important definitions/ results are enumerated here :

  • Inequalities involve a relation between two real numbers or algebraic expressions.
  • The inequality relations are "<", ">", "≤" and "≥".
  • Equal numbers can be added or subtracted to both sides of an inequality.
  • Both sides of an inequality can be multiplied or divided by a positive number without any change in the inequality relation.
  • Both sides of an inequality can be multiplied or divided by a negative number with reversal of inequality relation.
  • Both sides of an inequality can be squared, provided expressions are non-negative. As a matter of fact, this conclusion results from rule that we can multiply both sides with a positive number.
  • When both sides are replaced by their inverse, the inequality is reversed .

Equivalently, we may state above deductions symbolically.

If x > y , then :

x + a > y + a

a x > a y ; a > 0

a x < a y ; a < 0

x 2 > y 2 ; x , y > 0

1 x < 1 y ; when “x” and “y” have same sign.

It is evident that we can deduce similar conclusions with the remaining three inequality signs.

Intervals with inequalities

In general, a continuous interval is denoted with "less than (<)" or "less than equal to (≤)" inequalities like :

1 < x 5

The segment of a real number line from a particular number extending to plus infinity is denoted with “greater than” or “greater than equal to” inequalities like :

x 3

The segment of real number line from minus infinity to a certain number on real number line is denoted with “less than(<) or less than equal to (≤)” inequalities like :

x - 3

Two disjointed intervals are combined with “union” operator like :

1 < x 2 x > 5

Linear inequality

Linear function is a polynomial of degree 1. A linear inequality can be solved for intervals of valid “x” and “y” values, applying properties of inequality of addition, subtraction, multiplication and division. For illustration, we consider a logarithmic function, whose argument is a linear function in x.

f x = log e 3 x + 4

The argument of logarithmic function is a positive number. Hence,

x > - 4 3

Therefore, interval of x i.e. domain of logarithmic function is - 4 / 3, . The figure shows the values of “x” on a real number line as superimposed on x-axis. Note x= - 4/3 is excluded.

Graph of logarithmic function

Domain is traced on x-axis.

When f(x) = 0,

3 x + 4 = e f x = e 0 = 1

x = - 1

It means graph intersects x-axis at x=-1 as shown in the figure. From the figure, it is clear that range of function is real number set R.

We shall similarly consider inequalities involving polynomials of higher degree, rational function etc in separate modules.

Problem : A linear function is defined as f(x)=2x+2. Find valid intervals of “x” for each of four inequalities viz f(x)<0, f(x) ≤ 0, f(x)>0 and f(x) ≥ 0.

Solution : Here, given function is a linear function. At y=0,

f x = 2 x + 2 = 0

x = - 1

At x=0,

f x = 2

We draw a line passing through these two points as shown in the figure. From the figure, we conclude that :

Graph of linear function

Graph is continuous for all values of x.

f x < 0 ; x - , - 1

f x 0 ; x ( - , - 1 ]

f x > 0 ; x - 1,

f x 0 ; x [ - 1, )

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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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