The domain of the function $f\left(x\right) = sin^{-1}\left(\frac{x+5}{2}\right)$ is
- [-1,1]
- [2, 3]
- [3, 7]
- [-7, -3]
The Correct Option is D
Solution and Explanation
$\left[\because\right.$ Domain of $\sin ^{-1} x$ is $\left.x \in[-1,1]\right]$
$\therefore$ For $f(x)$ to be defined
$ -1 \leq \frac{x+5}{2} \leq 1 $
$ \Rightarrow -2 \leq x+5 \leq 2 $
$ \Rightarrow -2-5 \leq x \leq 2-5$
$ \Rightarrow -7 \leq x \leq-3 $
$ \therefore x \in[-7,-3] $
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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