Increasing and Decreasing Functions

Increasing and decreasing functions are special functional relational in mathematics whose graphs in a particular domain (which can be entire domain) exhibit a property of going upwards and downwards respectively  as we move along the the x-axis on the right-hand side. Intuitively functions can also be defined in contrasting nature of an increasing function can be called a non-decreasing function and the decreasing functions can be called as non-increasing functions. The derivative (representing tangential equation to curve) of any function is used when we wish to determine the intervals in which the function is increasing or decreasing. 

Some Basic Definitions:

Relation: A relation defined as R from a non-null set D is a subset which is obtained by the cartesian product C D. The said subset is obtained by mapping a mathematical relationship between the first and second in the ordered pairs of the cartesian product in C D.

Function: Functions are a special kind of relation. A relation  g from a set C to a non-null set D is identified as a function if each and every element from set C has only one image/mapping in set D. In other words, no two different elements from the non-null set D can possess the same preimage.

Domain: Domain belonging to the relation R can be defined as collection or the set of all first elements from the ordered pairs in a relation R from a non-null set C to a set D. It is also termed mathematically as a set of ‘inputs’ or ‘pre-images’.

Range: Range of the relation R can be defined as the set of all elements that are second in the ordered pairs for a relation R  defined from a non-null set C to a set C. It is also termed as the set of images. 

Co-domain: The whole of the set D in any relation R from a non-null set C  to a set D is termed as the codomain of that relation R. Range ⊆ Codomain

Strictly Increasing and Decreasing Functions 

1. Strictly Increasing Function – A mathematical function g(x) is said to be a function that increasing or non-decreasing in an particular interval if when evaluated for any two numbers c and d in such a way/manner that c<d, we have g(c) ≤ g(d). 

A function g(x) is termed as as strictly increasing or strictly non-decreasing in an particular interval  if when evaluated for any two numbers c  and d in such a way/manner that c<d, we have g(c) < g(d). All strictly increasing functions are increasing functions but on the contrary all increasing functions are not strictly increasing. 

The slope or the derivative of such a function is non-negative in the interval in which we are testing the function to be increasing or decreasing. 

2. Strictly Decreasing Function – A mathematical function g(x) is said to be a function that decreasing or non-increasing in an particular interval if when evaluated for any two numbers c and d in such a way/manner that c<d, we have g(c) g(d). 

A function g(x) is termed as as strictly decreasing or strictly non-increasing in an particular interval  if when evaluated for any two numbers c  and d in such a way/manner that c<d, we have g(c) > g(d). All strictly decreasing functions are decreasing functions but on the contrary all decreasing  functions are not strictly decreasing.

The slope or the derivative of such a function is non-positive in the interval in which we are testing the function to be increasing or decreasing. 

Algebraic Properties of  Increasing and Decreasing Functions 

1.If any two well defined functions r and s are increasing function (which is when evaluated for any two numbers c  and d in such a way/manner that c<d, we have g(c) g(d)) on any specific open defined interval, then the addition/sum of the functions r and s is also increasing in this particular interval

2. If any two well defined functions r and s are increasing function (which is when evaluated for any two numbers c and d in such a way/manner that c<d, we have g(c) g(d)) on any specific open defined interval, then the difference/subtraction of the functions r and s is also increasing in this particular interval.

3. If any two well defined functions r and s are decreasing function (which is when evaluated for any two numbers c and d in such a way/manner that c<d, we have g(c)g(d)) on any specific open defined interval, then the sum/addition of the functions r and s is also decreasing in this particular interval.

4. If any two well defined functions r and s are decreasing function (which is when evaluated for any two numbers c and d in such a way/manner that c<d, we have g(c)g(d)) on any specific open defined interval, then the difference/subtraction of the functions r and s is also decreasing in this particular interval.

Conclusion :

The increasing and decreasing functions find their special application in application of derivatives which is extrapolated in sciences to build on instruments/simulation that constantly decrease or increase in output by varying outputs. These functions are measured in relation to changes in independent variables whose value is arbitrarily or scientifically chosen by us to study parameters we are exploring. 

Increasing and decreasing functions are a kind or a property of some functions which makes them useful for us. It helps us determine the optimum interval of increase and decrease.