The domain of the function
\(f(x) = \sin^{-1}\left(\frac{x^2 - 3x + 2}{x^2 + 2x + 7}\right)\)
is :
The domain of the function
\(f(x) = \sin^{-1}\left(\frac{x^2 - 3x + 2}{x^2 + 2x + 7}\right)\)
is :
\([1,∞)\)
\([−1,2]\)
\([−1,∞)\)
\((−∞,2]\)
The Correct Option is C
Solution and Explanation
\(f(x) = \sin^{-1}\left(\frac{x^2 - 3x + 2}{x^2 + 2x + 7}\right)\)
\(-1 \leq \frac{x^2 - 3x + 2}{x^2 + 2x + 7} \leq 1\)
\(\frac{x^2 - 3x + 2x}{2 + 2x + 7} \leq 1\)
\(x^2−3x+2≤x^2+2x+7\)
\(5x≥−5\)
\(x≥−1 …(i)\)
\(\frac{x^2 - 3x + 2}{x^2 + 2x + 7} \geq -1\)
\(x^2−3x+2≥−x^2−2x−7\)
\(2x^2−x+9≥0\)
\(x∈R …(ii)\)
\((i)∩(ii)\)
\(Domain ∈ [−1,∞)\)
So, the correct option is (C): \([−1,∞)\)
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Concepts Used:
Types of Functions
Types of Functions
One to One Function
A function is said to be one to one function when f: A → B is One to One if for each element of A there is a distinct element of B.
Many to One Function
A function which maps two or more elements of A to the same element of set B is said to be many to one function. Two or more elements of A have the same image in B.
Onto Function
If there exists a function for which every element of set B there is (are) pre-image(s) in set A, it is Onto Function.
One – One and Onto Function
A function, f is One – One and Onto or Bijective if the function f is both One to One and Onto function.
Read More: Types of Functions