The domain of $ f(x) = sin^{-1} [Log_2 (\frac {x} {2})] $ is
- $0 \leq x \leq 1$
- $0 \leq x \leq 4$
- $1 \leq x \leq 4$
- $4 \leq x \leq 6$
The Correct Option is C
Solution and Explanation
Since, $-1 \leq \sin x \leq-1$
$\Rightarrow -1 \leq \log _{2}\left(\frac{x}{2}\right) \leq 1$
$\Rightarrow 2^{-1} \leq x / 2 \leq 2^{1}$
$\Rightarrow 2^{0} \leq x \leq 2^{2}$
$\Rightarrow 1 \leq x \leq 4$
Required domain is $x \in[1,4].$
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Concepts Used:
Functions
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.
Kinds of Functions
The different types of functions are -
One to One Function: When elements of set A have a separate component of set B, we can determine that it is a one-to-one function. Besides, you can also call it injective.
Many to One Function: As the name suggests, here more than two elements in set A are mapped with one element in set B.
Moreover, if it happens that all the elements in set B have pre-images in set A, it is called an onto function or surjective function.
Also, if a function is both one-to-one and onto function, it is known as a bijective. This means, that all the elements of A are mapped with separate elements in B, and A holds a pre-image of elements of B.
Read More: Relations and Functions