Functions in Math

Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x. Mapping or transformation is used to denote a function in math. These functions are usually denoted by letters such as f, g, and h. The domain is defined as the set of all the values that the function can input while it can be defined. The range is all the values that come out as the output of the function involved. Co-domain is the set of values that have the potential of coming out as outputs of a function. Let us explore the world of functions in math.

What Are Functions?

A function is a process or a relation that associates each element ‘a’ of a non-empty set A , at least to a single element ‘b’ of another non-empty set B. A relation f from a set A (the domain of the function) to another set B (the co-domain of the function) is called a function in math. f = {(a,b)| for all a ∈ A, b ∈ B}

• A relation is said to be a function if every element of set A has one and only one image in set B.
• A function is a relation from a non-empty set B such that the domain of a function is A and no two distinct ordered pairs in f have the same first element.
• A function from A → B and (a,b) ∈ f, then f(a) = b, where ‘b’ is the image of ‘a’ under ‘f’ and ‘a’ is the preimage of ‘b’ under ‘f’.
• If there exists a function f: A → B, the set A is called the domain of the function f, and the set B is called its co-domain.

Examples of Functions

Whenever we say that a variable quantity y is a function of a variable quantity x, we mean to say that: y depends on x; the value of y is determined by the value of x. We can write this dependence as follows: y = f (x)

1. The area of a circle can be expressed in terms of its radius A = π r². Area A is dependent on the radius r. In the language of functions in math, we say that A is a function of r.
2. The volume V of a sphere is a function of its radius. The dependence of V on r is given by V = 4/3 π r³.
3. The acceleration a of a body of fixed mass m is a function of the force F applied on the body: a = F/m.
4. The power P dissipated in a resistor of fixed resistance R is a function of the current I passing through the resistor: P=I2RP=I2R.
5. Suppose that a taxi charges a down payment of $10, and subsequently, $ 2 for every mile traveled. The taxi fare F is a function of the distance d traveled, and the dependence between the two quantities is given by F = 10 + 2d (F is in dollars and d is in miles).