Relations and Functions

Let A = {9, 10, 11, 12, 13} and let f : A → N be defined by f(n) = the highest prime factor of n. Find the range off.

Let f be the subset of Z x Z defined by f = {(ab, a+ b): a, b ∈ Z}. Is f a function from Z to Z: justify your answer.

Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true?

(i) f is a relation from A to B.

(ii) f is a function from A to B.

Justify your answer in each case.

Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b²}. Are the following true?

(i) (a, a) ∈ R, for all a ∈ N
(ii) (a, b) ∈ R, implies (b, a) ∈ R

(iii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R.

Justify your answer in each case.

Let f = {(1, 1), (2, 3), (0, -1), (-1, -3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.

Letbe a function from R into R. Determine the range of f.

Find the domain and the range of the real function f defined by f(x) = |x- 1|.

The relation f is defined by 

The relation g is defined by 

Show that f is a function and g is not a function.

Find the domain of the function 

If f(x) =  x², find

Find the range of each of the following functions.

(i) f(x) = 2 - 3x, x ∈ R, x> 0.

(ii) f(x) = x²+ 2, x, is a real number.

(iii) f(x) = x, x is a real number

A function fis defined by f(x) = 2x- 5. Write down the values of

(i) f(0), (ii) f(7), (iii) f(-3)

Find the domain and range of the following real function:

(i) f(x) = -|x| (ii) 

Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

(iii) {(1, 3), (1, 5), (2, 5)}

Let R be the relation on Zdefined by R = {(a, b): a, b ∈ Z, a - b is an integer}. Find the domain and range of R.

Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.

 Write the relation R = {(x, x³): x is a prime number less than 10} in roster form.

Determine the domain and range of the relation R defined by R = {(x, x+ 5): x ∈{0, 1, 2, 3, 4, 5}}.

Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by

{(a, b): a, b ∈A, b is exactly divisible by a}.

(i) Write R in roster form

(ii) Find the domain of R

(iii) Find the range of R.

The given figure shows a relationship between the sets P and Q. write this relation

(i) in set-builder form (ii) in roster form.

What is its domain and range?

A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between xand yis odd; x ∈A, y ∈B}. Write R in roster form.

Define a relation R on the set Nof natural numbers by R = {(x, y): y= x+ 5, xis a natural number less than 4; x,y ∈ N}. Depict this relationship using roster form. Write down the domain and the range.

Let A = {1, 2, 3, ... , 14}. Define a relation R from A to A by R = {(x, y): 3x - y = 0, where x, y ∈ A}. Write down its domain, codomain and range.

The Cartesian product A x A has 9 elements among which are found (-1, 0) and (0, 1). Find the set A and the remaining elements of A x A.

Let A and B be two sets such that n(A) = 3 and n (B) = 2. If (x, 1), (y, 2), (z, 1) are in A x B, find A and B, wherex, y and z are distinct elements.

Let A = {1, 2} and B = {3, 4}. Write A x B. How many subsets will A x B have? List them.

 Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that

(i) A x (B ∩C) = (A x B) ∩ (A x C)

(ii) A x C is a subset of B x D

If A x B = {(a, x), (a, y), (b, x), (b, y)}. Find A and B.

If A = {-1, 1}, find A x A x A

State whether each of the following statement are true or false. If the statement is false, rewrite the given statement correctly.

(i) If P = {m, n} and Q = {n, m}, then P x Q = {(m, n), (n, m)}.

(ii) If A and B are non-empty sets, then A x B is a non-empty set of ordered pairs (x, y) such that x ∈ A and y ∈ B.

(iii) If A = {1, 2}, B = {3, 4}, then A x (B ∩ Φ) = Φ.

If G = {7, 8} and H = {5, 4, 2}, find G x H and H x G.

If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A x B)?

If, find the values of xand y.

Define polynomial function. Draw the graph of f (x) = x³ find domain and

range

Draw the graphs of the following real functions and hence find range:

f (x) = x²