If f(x) is a real valued function, then which of the following is injective function ?
(i) `f(x) = x^(5)` (ii) `f(x) = x + sin x`
(iii) `f(x) = x^(2)` (iv) `f(x) = e^(x)`
(v) `f(x) = x^(3) + x^(2) + 4x + 4`

Explanation

(i) `f'(x) = 5x^(4) ge 0` for all `x in R`
Hence, it is one-one.
(ii) `f'(x) = 1 + cos x ge 0`
Hence, f(x) is one-one.
(iii) f'(x) = 2x may be positive or negative, hence it is not one-one.
(iv) `f'(x) = e^(x) gt 0`, hence it is one-one.
(v) `f'(x) = 3x^(2) + 2x + 4 = 3 (x + (1)/(3))^(2) + (11)/(3) gt 0`.
Hence, f(x) is one-one.