Composition of functions

Suppose A, B and C are sets and $f:A \rightarrow B$ and $g:B \rightarrow C$ are functions. We define a function

$h: A \rightarrow C$ by

$h(a)=g(f(a))$ for every $a \in A$.

It is easily seen that h is a well-defined function, as $f(a) \in B$ for every $a \in A$ and $g(b) \in C$ for every $b \in B$. The function h is called the composition of the functions g and f and is denoted by $g \circ f$.

Examples.

a) Suppose that in a forest, carnivorous animals sustain themselves by feeding only on herbivorous animals and the nutrition level of a herbivorous animal depends on the vegetation around the animal. So the nutrition level of a carnivorous animal ultimately depends on the vegetation around the population of herbivorous animals it feeds on. Thus, if V is the density of vegetation around the herbivorous animals n is the level of nutrition of the herbivorous animal (for simplicity measured by its weight, though there are often more parameters depicting the level of nutrition of an animal), $n: V \rightarrow \Re$ is a function. Similarly, if c is the level of nutrition of a carnivorous animal, then c is a function depending on the level of nutrition of herbivorous animals it feeds on:

$c: \Re \rightarrow \Re$

Thus, $c \circ n: V \rightarrow \Re$

is the level of nutrition of the carnivorous animal ultimately depending on the density of vegetation.

b) Take for example the force experienced by a moving charged particle in a magnetic field which is varying in time. We know that the force on the charged particle depends on the strength of the magnetic field in which it moves. Again, as the magnetic field strength varies with time, the force experienced is ultimately a function of time and position.

c) Suppose there is a lamp  in the room. The intensity of illumination at point in the room depends on the illuminating power of the lamp. The illuminating power of the lamp again depends on the voltage of the electricity supply which makes the lamp glow. So, ultimately, the intensity of illumination depends on the voltage of the power supply.

Exercise.

Give five more examples of composition of functions.

More later,

Nalin Pithwa

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