Total number of rational values
How many rational values of a exist for which the range of the function y = 1 − x 2 − a x − 1 does not contain any value from the interval [ − 1 , 1 ] .
Enter -1 as your answer if your answer comes out to be infinite.
The answer is 0.
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1 solution
What about y at a = 0 ?
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It would be undefined if we put x = 1 ⟹ value of y will not exist.
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When a = 0 , as x → 1 , y → 2 − 1 . For all values of x > 1 , y ∈ ( 2 − 1 , 0 ) . So the answer should be 1 since there exists only one value of a that makes y to lie in [ − 1 , 1 ] .
at ( a = 0 ) ⇒ y = 1 − x 2 x − 1 ⇒ y = − ( x − 1 ) ( x + 1 ) x − 1 ⇒ y = − x + 1 1
Putting ( x = 1 ) ⇒ y = − 2 1 ∈ [ − 1 , 1 ]
Putting x = 1 , the value of y comes out to be 0.
No matter what the value of a be , at x = 1 , value of y will be zero and between [ − 1 , 1 ] .
Hence there is no possible value of a .