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Let f : (0, oo) rarr [9, oo) defined as ...

Let `f : (0, oo) rarr [9, oo)` defined as `f(x) = x^(12) + (4)/(x^(2)) + (4)/(x)`. Check whether f is onto or not.

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`f : (0, oo) rarr [9, oo)`
`f(x) = x^(12) + (4)/(x^(2)) + (4)/(x)`
Applying A.M., G.M. inequality
`(x^(12) + (1)/(x^(2)) + (1)/(x^(2))+ (1)/(x^(2))+(1)/(x) + (1)/(x) + (1)/(x) + (1)/(x))/(9) ge (x^(12).((1)/(x^(2)))^(4)((1)/(4))^(4))^(1//9)`
`x^(12) + (4)/(x^(2)) + (4)/(x) ge 9` i.e., Range of is `[9, oo) = 0` co-domain of f
Hence, f is onto function.
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