Common Math Functions

The common math function is a one of the major type of relation. In a function, number two ordered pairs can have the same first element and a different second element. That is, for 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.The set of all images of elements of A under f is called the range of f. The common math functions example problems and practice problems are given below.

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Example problems for common math functions:

Example problem 1:

Let A = Z – { 0 } and let f : A  →A be defined by f(x) = ` |x|/x` . What type of function is f?

Solution:

By definition, A = {+1, +2, +3, …}

f(x) =` |x|/x`

f(1) = `1/1` = -1 and

f(-1) = `1/-1` = -1

f(-2) = `2/2 ` = 1 and

f(-2) = `2/-2 ` = -1

Thus for any a ∈ Z – { 0 }, f(a) = 1 and f(–a) = -1. ∴ The range of f is {–1, 1}. Hence f is into and many to one function.

Example problem 2:

Which of the following relations from A = {5, 6, 7, 8} to B = { l , m, n, p} are functions.

(a) {(5, l ), (6, l ), (7, m), (8, p)}

(b) {(6, l ), (7, p), (8, n)}

Solution:

(a) All the elements in A occur as first elements in the ordered pairs in f and no element in A is repeated (Fig). Hence f is a function

Arrow diagram


(b) Here f is not a function because 5 ∈ A is not associated with an element of B

Arrow diagram

Practice problems for common math functions:

Practice problem 1:

Which of the following relations are functions from A = {1, 4, 9, 16} to B = {–1, 2, –3,–4}

(a) f = {(1, –1), (4,2), (9, –3), (16, –4)}

(b) f = {(1, –4) (1, –1), (9, –3), (16, 2)]

Answer: (a) function (b) not a function

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Practice problem 2:

X = {–4, –2, 0, 2, 4}, Y = {0, 1, 4, 9, 16} and f : X → Y is defined by f(x) = x2. Check whether f is function

Answer: yes, it is a function