Using Monotonicity to Find the Range of a Function
fx =|x+5|-2/x...
Question
f(x)={|x+5|−2x2+10x+21},where {.} denotes fractional part function then which of the following is true?
A
Number of integers in the domain of f(x) is 2.
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B
Number of integers in the domain of f(x) is 3.
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C
Range of f(x) is (0,12]
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D
Number of integers in the domain of f(x) is infinite.
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Solution
The correct option is D Number of integers in the domain of f(x) is infinite. We have, f(x)={|x+5|−2x2+10x+21}
Let |x+5|=t f(x)={t−2t2−4}={(t−2)(t−2)(t+2)} ={1|x+5|+2}
As t≠2 ⇒|x+5|≠2 ⇒x+5≠±2 ⇒x≠−3,−7
domain is x∈R−{−3,−7}
Range =[0,12]−{14}