# Functions-18

### From Homeworkwiki

**Find whether the following functions are one-one or not:**

**(i) f : R → R given by f(x) = x ^{3} + 2 for all x ∈ R.**

**(ii) f : Z → Z given by f(x) = x**

^{2}+ 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) => x^{3} + 2 = y^{3} + 2

=>x^{3} = y^{3} => 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) => x^{2} + 1 = y^{2} + 1 => x^{2} = y^{2}

=>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) = 2^{2} + 1 = 5 and f(-2) = (-2)^{2} + 1 = 5.

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