Functions-18

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Find whether the following functions are one-one or not:
(i) f : R → R given by f(x) = x3 + 2 for all x ∈ R.
(ii) f : Z → Z given by f(x) = x2 + 1 for all x ∈ Z

Solution: Let x, y be two arbitrary elements of R (domain of f) such that f(x) = f(y). Then,

f(x) = f(y) => x3 + 2 = y3 + 2

=>x3 = y3 => x = y

Hence, f is a one-one function from R to itself.

(ii) Let x, y be two arbitrary elements of Z such that f(x) = f(y). Then,

f(x) = f(y) => x2 + 1 = y2 + 1 => x2 = y2

=>x = ± y

Here, f(x) = f(y) does not provide the unique solution x = y but it provides x = ± y. So, f is not a one-one function. In fact,

f(2) = 22 + 1 = 5 and f(-2) = (-2)2 + 1 = 5.

So, 2 and -2 are two distinct elements having the same image.

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