Let f : N → Y be a function defined as f (x) = 4x + 3, where Y = {y ∈ N : y = 4x + 3 for some x ∈ N}.Show that f is invertible and its inverse is 

Let R be the real line. Consider the following subsets of the plane R × R.
S = {(x, y) : y = x + 1 and 0 < x < 2}, T = {(x, y) : x − y is an integer}. Which one of the following is true?

• neither S nor T is an equivalence relation on R

• both S and T are equivalence relations on R

• S is an equivalence relation on R but T is not 

• S is an equivalence relation on R but T is not 

Let f(x) = Then which one of the following is true?

• f is neither differentiable at x = 0 nor at x = 1

• f is differentiable at x = 0 and at x = 1

• f is differentiable at x = 0 but not at x = 1 

• f is differentiable at x = 0 but not at x = 1 

The largest interval lying in  for which the function  is defined, is

• [0, π]

• 

• [-π/4, π/2)

• [-π/4, π/2)

155.

Let f : R → R be a function defined by f(x) = Min {x + 1, |x| + 1}. Then which of the following is true ?

• f(x) ≥ 1 for all x ∈ R

• f(x) is not differentiable at x = 1

• f(x) is differentiable everywhere

• f(x) is differentiable everywhere

The function f: R ~ {0} → R given by

can be made continuous at x = 0 by defining f(0) as

• 2

• -1

• 1

• 1

The number of values of x in the interval [0, 3π] satisfying the equation 2sin² x + 5sinx − 3 = 0 is

• 4

• 6

• 1

• 1

The set of points where x f(x) = x /1+|x| is differentiable is

• (−∞, 0) ∪ (0, ∞)

• (−∞, −1) ∪ (−1, ∞)

• (−∞, ∞)

• (−∞, ∞)

Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12} be a relation on the set A = {3, 6, 9, 12}. The relation is

• reflexive and transitive only

• reflexive only

• an equivalence relation

• an equivalence relation

Let f : (-1, 1) → B, be a function defined by  then f is both one-one and onto when B is the interval

• 

• [0, π/2)

• 
•