If in a population of size ”N” the birth rate is represented as “b” and the death rate as “d”, the increase or decrease in “N” during a unit time period ”t” will be: The equation given above can also be represented as: What does ”r” represent. Write any one significance of calculating ”r” for any population?

r here represents the intrinsic rate of increase of population and it is equal to the difference in the natality (birth rate) and mortality (death rate). The 'r' is called the 'intrinsic rate of natural increase. It' is a very important parameter chosen for assessing impacts of any biotic or abiotic factor on population growth. If the value of 'r' is increasing it signifies the growth of the population and if it is going down then the population growth has slowed down.

Show that the function f : R → R defined by f x = x x 2 + 1 , ∀ x ∈ ℝ is neither one-one nor onto. Also, if g : R → R is defined as g ( x ) = 2 x - 1 , find fog ( x ) .

Given, y = x x 2 + 1 ⇒ y x 2 − x + y = 0 Here a = y , b = - 1 and c = y ∴ x = − ( − 1 ) ± 1 − 4 y 2 2 y Clearly, for every value of y , x will have two different values . So the function is many-one not one - one. Since 1 − 4 y 2 ≥ 0 ⇒ ( 1 + 2 y ) ( 1 − 2 y ) ≥ 0 ⇒ − 1 2 ≤ y ≤ 1 2 That means no matter what is x , y belongs to the interval − 1 2 , 1 2 . Hence, the function is not onto. Now, f o g x = f g x = f 2 x - 1 = 2 x - 1 2 x - 1 2 + 1 = 2 x + 1 4 x 2 - 4 x + 1 + 1 = 2 x + 1 2 2 x 2 - 2 x + 1

If A = 1 , 2 , 3 , show that a onto function f : A → A must be one-one.

Given A = 1 , 2 , 3 Possible onto functions from A to A can be the following: ( i ) ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) ( i i ) ( 1 , 1 ) , ( 2 , 3 ) , ( 3 , 2 ) ( i i i ) ( 1 , 2 ) , ( 2 , 2 ) , ( 3 , 3 ) ( i v ) ( 1 , 2 ) , ( 2 , 1 ) , ( 3 , 3 ) ( v ) ( 1 , 3 ) , ( 2 , 2 ) , ( 3 , 1 ) ( v i ) ( 1 , 3 ) , ( 2 , 1 ) , ( 3 , 2 ) Here, in each function, different elements of the domain have different images. So, all the functions are one-one.

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Which thematic map is best for representing population density?

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Important Questions on Relations and Functions

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The principle of competitive exclusion was stated by

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Mathematics>Algebra>Relations and Functions>Types of Functions

If in a population of size ”N” the birth rate is represented as “b” and the death rate as “d”, the increase or decrease in “N” during a unit time period ”t” will be:

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