What are Functions in Math

Introduction for functions in math:

A math function is a one of the type of relation. In a function, number 2 ordered pairs can have the same first element and a different second element. That is, for a functions, corresponding to every 1st element of the ordered pairs, there must be a different 2nd element. i.e. In a function we can't have ordered pairs of the form (a1, b1) and (a2, b2) with a1 = a2 and b1 ? b2. It is called as function. Types of functions and example problems are given below.

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Types of functions in math:


Let us see about some important math functions types are given below.

1. Into function:

A function f: A ? B is said to be into function if its range is a proper subset of its co-domain.

Let f: A ? B, where A = {a, b, c}, B = {p, q, r, s} be defined by f(a) = p, f(b) = q, f(c) = r. Then the range of f is {p, q, r}, a proper subset of B `=>` f is an into function.

Let g: C ? D where C = {6, 1, 3, 4} and D = {m, n, p} be defined by g(6) = p, g(1) = p, g(3) = p and g(4) = p. Then g is into as its range {p} is a proper subset of D `=>` g is an into function.

2. Onto function:

A function from A to B is said to be an onto function if its range is equal to its co-domain B. In other words, f is called ‘onto’ if every element of B has a Preimage in A.

For example:

(i) Let A = {1, 3, 4, 5} and B = {3, 6,7}, f(1) = 6, f(3) = 3, f(4) = 7 and f(5) = 6. Evidently the range of f is B and is onto.

(ii) Consider g : C ? D where C = {3, –3, 4, –4} and D = {9, 16} and g(3) = 9, g(–3) = 9, g(4) = 16, g(–4) = 16. Evidently g is onto.

3. One to one function:

A function f from A to B is called an one to one function (one – one function) if for a, b ? A, where a ? b, we must have f(a) ? f(b). Here the domain and the range of f will have the same cardinality.

For example:

(i) Let f : A ? B where A = {a, b, c}, B = {–1, 2, 3, 5} f(a) = 5, f(b) = –1, f(c) = 3. Then f is one to one. Note that the range of f is a subset of B and so f is into. We call such functions as one to one into.

(ii) Let g: C ? D where C = {3, 5, 7} and D = {4, 6, 8} be defined by g(3) = 4, g(5) = 6, g(7) = 8. Then g is one to one and the range of g is equal to the co domain D. So g is onto. We call such functions as one to one onto.

4. Constant function:

A function f from A to B is called a constant function if every element of A has the same image in B.

Let A = {4, 5, 6, 7}, B = {7, 8, 10} and if f : A ? B is defined by f(4) = 10, f(5) = 10, f(6) = 10 and f(7) = 10 then f is constant function. I have recently faced lot of problem while learning algebra 2 math help, But thank to online resources of math which helped me to learn myself easily on net.


Example problem for functions in math:


Example problem:

Given A = {a, b, c, d}, B = {p, q, r, s} and if f : A ? B is defined as below find the type of the function:

(i) f = {(a, p), (b, p), (c, q), (d, r)}

(ii) f = {(a, q), (b, q), (c, q), (d, q)}

(iii) f = {(a, p), (b, q), (c, r), (d, s)}

Solution:

(i) Since a, b are mapped to p ? B and the element s of B has no Preimage in A, it is many to one and into function

(ii) All the elements of A are mapped to the similar element q ? B. Hence it is a constant function. It is also many to one.

(iii) Because all the images are different and no element in B is left out without any Preimage, f represents a one to one and onto function.