Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1}
| List-I | List-II | ||
|---|---|---|---|
| (A) | $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ | (I) | $[\frac{1}{3} , 1 ]$ |
| (B) | $(x)=|[x]|,x \in [R$ | (II) | Z |
| (C) | $h(x) = |x - [x]| , x \in [R$ | (III) | W |
| (D) | $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ | (IV) | [0, 1) |
| (V) | { -1, 1} | ||
- (A)-(V), (B)-(III) , (C)-(II), (D)-(I)
- (A)-(III), (B)-(II) , (C)-(IV), (D)-(I)
- (A)-(V), (B)-(III) , (C)-(IV), (D)-(I)
- (A)-(I), (B)-(II), (C)-(III), (D)-(IV)
The Correct Option is C
Solution and Explanation
So, range of $f(x)$ is $\{-1,1\}$.
(B) $\because g(x)=|[x]|, x \in R$
As $[x] \in I \Rightarrow|[x]| \in W$
So, range of $g(x)$ is $W$.
(C) $\because h(x)=|x-[x]|, x \in R=|\{x\}| \in[0,1)$
$[\because\{x\}=x-[x]$ and $\{x\} \in[0,1)]$
So, range of $h(x)$ is $[0,1)$.
(D) $\because f(x)=\frac{1}{2-\sin 3 x}, x \in R$
$\because -1 \leq \sin 3 x \leq 1, \forall x \in R$
$\Rightarrow -1 \leq-\sin 3 x \leq 1$
$\Rightarrow 2-1 \leq 2-\sin 3 x \leq 2+1$
$\Rightarrow \frac{1}{3} \leq \frac{1}{2-\sin 3 x} \leq \frac{1}{1}$
So, range of $f(x) \text { is }\left[\frac{1}{3}, 1\right]$
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Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
