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Range = [ f x 1 , f x 2 ]

We shall study this aspect of finding range in detail in a separate module.

Non-decreasing function or increasing

The successive value of function increases or remains constant as the value of the independent variable increases. In other words, the preceding values are less than or equal to successive values that follow. Mathematically,

If x 1 < x 2 then f x 1 f x 2

Non decreasing function

The successive function value is greater than or equal to previous value.

As f( x 1 )≤f( x 2 ) for all x 1 , x 2 ∈X, the difference “f(x+h) – f(x)” is non-negative for “h”, however small. This implies that the first derivative of function is non-negative. If we think of possibility, then we can realize that tangent to the function curve can be parallel to x-axis for a subset of X, while curve is increasing overall in the interval. It means that first derivative can be equal to zero points or sub-intervals in which it is increasing. Thus, for non-decreasing function,

f x 0 ; Equality sign holds for few points or a continuous section in X

For increasing function, if x 1 < x 2 , then f( x 1 ) ≤ f( x 2 ), for all x 1 , x 2 ∈X. This means that there may be same function values for different values of x. This is “many one” relation and as such function is not invertible in X.

Strictly decreasing function

The successive value of function decreases as the value of the independent variable increases. In other words, the preceding values are greater than successive values that follow. Mathematically,

If x 1 < x 2 then f x 1 > f x 2

Strictly decreasing function

The successive function value is less than previous value.

Problem : Determine monotonic nature of the function in the interval (-∞,0].

y = x 2

Solution : Let x 1 and x 2 belong to the interval [0,∞) such that x 1 < x 2 . Multiplying inequality with x 1 (a negative number) changes the nature of inequality :

x 1 2 > x 1 x 2

Multiplying inequality with x 2 (a negative number) changes the nature of inequality :

x 1 x 2 > x 2 2

Combining two inequalities,

x 1 2 > x 2 2 f x 1 > f x 2

Thus, given function is strictly decreasing in (-∞,0].

As f( x 1 )>f( x 2 ) for all x 1 , x 2 ∈X, the difference “f(x+h) – f(x)” is negative for “h”, however small. This implies that the first derivative of function is negative. If we think of possibility, then we can realize that tangent to the function curve can be parallel to x-axis for couple of x values, while curve is continuously decreasing in the interval. It means that first derivative can be equal to zero for few points in the interval in which it is strictly decreasing. Thus, for strictly decreasing function,

f x 0 ; Equality sign holds for points only - not on a continuous section in X

For strictly decreasing function, if x 1 < x 2 , then f( x 1 )>f( x 2 ), for all x 1 , x 2 ∈X. It means that all distinct x values correspond to distinct y values and vice-versa. Therefore, strictly decreasing function is one-one function i.e. a bijection and hence “invertible”. In other words, if a function has strict decreasing order, then it is invertible. Mathematically, we say that if f’(x) ≤ 0 (equality holding for points only); x∈X, then function is invertible in X. For example, consider sine function,

f x = sin x f x = cos x

We know that cosx is negative in the interval [π/2, 3π/2]. Hence sine function is a strictly decreasing function in [π/2, 3π/2]and is invertible. Recall though that inverse sine function is not defined in this interval, but in basic interval about origin [-π/2,π/2].

The order of a function provides an easy technique to determine range of a continuous function, corresponding to a given domain interval. For example, if domain of a continuously decreasing function, f(x), is [ x 1 , x 2 ], then the least value of the function is f( x 2 ) and greatest value of the function is f( x 1 ). Hence, range of the function is :

Range = [ f x 2 , f x 1 ]

We shall study this aspect of finding range in detail in a separate module.

Non-increasing function or decreasing

The successive value of function decreases or remains constant as the value of the independent variable increases. In other words, the preceding values are greater than or equal to successive values that follow. Mathematically,

If x 1 < x 2 then f x 1 f x 2

Non-increasing function or decreasing

The successive function value is less than or equal to previous value.

As f( x 1 )≥ f( x 2 ) for all x 1 , x 2 ∈X, the difference “f(x+h) – f(x)” is non-positive for “h”, however small. This implies that the first derivative of function is non-positive. If we think of possibility, then we can realize that tangent to the function curve can be parallel to x-axis for a subset of X, while curve is decreasing overall in the interval. It means that first derivative can be equal to zero at points or in sub-intervals in which it is decreasing. Thus, for non-decreasing function,

f x 0 ; Equality sign holds for few points or a continuous section in X

For decreasing function, if x 1 < x 2 , then f( x 1 ) ≥ f( x 2 ), for all x 1 , x 2 ∈X. This means that there may be same function values for different values of x. This is “many one” relation and as such function is not invertible in X.

Questions & Answers

what is Economics
Ebem Reply
the branch of knowledge concerned with the production, consumption, and transfer of wealth and has Influence by sociology!!!!
Ajay
Economics is the study of how humans make decisions when they want to fulfil their requirements and desires for goods, services and resources.
Abdullah
Economics is the study how humans make decisions in the faces of scarcity.
Rose
economic is the study of how human make decision in the fact of scarcity.
Toang
Economics is a social science which study human behavior as a relationship between earn and scarce mean which have alternative uses
Juliet
what is market structure
Fatima
market structure in economics depicts how firms are differentiated and categorised based on types of goods they sell and how their operations are affected by external factors and elements.
Nasir
what is demand
Gooluck Reply
demand is the willingness to purchase something
Mohamed
demand is the potential ability or williness to purchases something at a particular price at a given period of time..
Ahmed
Demand refers to as quantities of a goods and services in which consumers are willing and able to purchase at a given period of time. Demand can also be defined as the desire backed by ability to purchase .
Fadiga
what is demand
John Reply
is the production of goods in scarcity
David
thanks
John
Demand refers to as quantities of a goods and services in which consumers are willing and able to purchase at a given period of time.
Fadiga
what is demand of supply
music Reply
What is the meaning of supply of labour
Anthonia Reply
what is production?
Elizabeth Reply
Production is basically the creation of goods and services to satisfy human wants
Anthonia
under what condition will demand curve slope upward from left to right instead of normally sloping downward from left to right
Atama Reply
how i can calculate elasticity?
Tewekel Reply
What is real wages
Emmanuella Reply
what are the concept of cost
Tabitha Reply
what is the difference between want and choice
Grace Reply
Want is a desire to have something while choice is the ability to select or choose a perticular good or services you desire to have at a perticular point in time.
Dalton
substitutes and complements
Amman Reply
Substitute are goods that can replace another good but complements goods that can be combined together
nkanyiso
account for persistent increase in lnflation
niwahereza Reply
what is opportunity cost
Adebowale Reply
opportunity cost reffered to as alternative foregone when choice is made
niwahereza
government measures to control inflation?
Formu Reply
control populationk growth rate by using family planning to reduce faster increase of people than job creation
niwahereza
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Beverly Reply
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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