by the rule
, for each real number x.
(i) Is H one-to-one? Prove or give a counterexample.
(ii) 1s H onto? Prove or give a counterexample.
b. Define by the rule , for each nonnegative real number x. Is K onto? Prove or give a counterexample.
To prove is one-to-one and onto otherwise give counterexample.
Define by the rule, for each real number .
Let us suppose and assume that and now prove
So from the definition of , the expression can be written as follows-
This shows that the values of domain will be different even if outputs are samebecause the square of any negative integer is also positive.
Thus two different values of when substituted into the function may give the same value.
Since the same output is possible for two different inputs so it is proved that where is not one-to-one.
For the function to be onto there must be some real value in the codomain for which there should be some real integer in the domain of the function
To prove is onto otherwise give counterexample.
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