If the function \( f: \mathbb{N} \to \mathbb{N} \) is defined as \( f(n) = \begin{cases} n - 1, & \text{if } n \text{ is even} \\ n + 1, & \text{if } n \text{ is odd} \end{cases} \) then: (A) \( f \) is injective (B) f is into (C) f is surjective (D) f is invertible Choose the correct answer from the options given below :
(A),(C) and (D) only
Let's analyze whether the given function \( f: \mathbb{N} \to \mathbb{N} \), defined as:
\(f(n) = \begin{cases} n - 1, & \text{if } n \text{ is even} \\ n + 1, & \text{if } n \text{ is odd} \end{cases}\)
possesses injective, into, surjective, or invertible properties. Here's how we will approach it:
(A) Injective?
A function is injective if different inputs give different outputs.
Here, each number maps to a unique partner, and no two different inputs give the same output.
Injective
(C) Surjective?
A function is surjective if every natural number is hit.
From the pattern:
- Every even number comes from the previous odd
- Every odd number comes from the next even
So every natural number has a preimage.
Surjective
(D) Invertible?
A function is invertible if it is both injective and surjective.
Since both hold,
Invertible
(In fact, the inverse is the function itself: \(f(f(n))=nf(f(n)) \)
(B) Into?
“Into” means not onto (not surjective). But we already showed it is surjective.
Not into
(A), (C), and (D) only, indicating that f is injective, into, and invertible under specific restrictions, making the reasoning and answer logical and calculated.
Important Questions from Relations and Functions
Which one of the following represents the correct feasible region determined by the following constraints of an LPP?
\( x + y \geq 10, \quad 2x + 2y \leq 25, \quad x \geq 0, \quad y \geq 0 \)
Let [x] denote the greatest integer function. Then match List-I with List-II:
List-I List-II (A) |x - 1| + |x - 2| (I) is differentiable everywhere except at x = 0 (B) x - |x| (II) is continuous everywhere (C) x - [x] (III) is not differentiable at x = 1 (D) x |x| (IV) is differentiable at x = 1 Choose the correct answer from the options given below:
Let \( R \) be the relation on \( \mathbb{N} \) (set of natural numbers) defined by \( R = \{(a, b) : a, b \in \mathbb{N} \) and \( b \) is divisible by \( a\} \). Then the relation \( R \) is:
If \( f: \mathbb{R} \to \mathbb{R} \) is a function given by \( f(x) = \lfloor x \rfloor \) (greatest integer function), then which of the following is/are correct?
- (a) \( f \) is one-one
- (b) \( f \) is not onto
- (c) Range of \( f \) is \( \mathbb{I} \) (set of integers)
- (d) \( f(2.5) = 2 \)
- (e) \( f \) is bijective
Choose the correct answer from the options given below:
If \( m \) and \( M \) are respectively minimum and maximum values of \( f(x) = |2 - |x|| \), for \( -3 \leq x \leq 3 \), then:
