Question:
If $f: R \rightarrow S $ defined by $ f(x)=\sin x-\sqrt{3} \,\cos\,x+\,$ 1, is onto, function then S = ?
If $f: R \rightarrow S $ defined by $ f(x)=\sin x-\sqrt{3} \,\cos\,x+\,$ 1, is onto, function then S = ?
Updated On: Jul 5, 2022
- [0, 3]
- [-1, 1]
- [0, 1]
- [-1, 3]
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The Correct Option is D
Solution and Explanation
Since $- 2 \leq \,\sin \,x -\sqrt{3} \,\cos\, x\, \leq\, 2$
$\therefore -1 < \sin x - \sqrt{3} \cos\, x \,+ 1 \leq 3$
$\therefore R_f = [-1, 3]$
$\therefore$ [-1, 3] holds
. $\begin{Bmatrix}
\because \sin \, x - \sqrt{3} \,\cos \, x \\[0.3em]
= 2\left[\frac{1}{2}\sin \, x= \frac{\sqrt 3}{2}\cos \, x\right] \\[0.3em]
= 2\left[\sin\left(x- \frac{\pi}{6} \right)\right]
\end{Bmatrix}$
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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.
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