225 A3 Q 1 For each of the following functions state wheth

225 - A3 - Q 1

For each of the following functions state whether it is injective and whether it is subjective. Briefly explain your answers. (Have you tried typing a function expression into Google?) f:R right arrow R where f(x) = x^3. f:[0,pi/2] right arrow R where f(x) = sin x. f: R right arrow R where f(x) = |x|. f: (0,1] right arrow [-1,1] where f(x) = sin (1-x).

Solution

Suppose, every function is extending from A -> B :

Injective means that every member of \"A\" has its own unique matching member in \"B\".

Surjective means that every \"B\" has at least one matching \"A\" (maybe more than one).

Now, in a) It is a continuous function extending over -infinity to +infinity, so It is a surjective as for every value from -infinity to + infinity of x³, it has at least one value of x corresponding to that.
Also, it is injective, since for every value of x it has one unique value for x³ corresponsing to that.

b) sinx goes from -1 to 1 and for x = [0,pi/2], sinx goes from 0 to 1. So, the function is not surjective as in the question the set \'B\' is mentioned as R, while we won\'t get any value in set A for values of R - [0,1] in set B.
Also, it is injective as for every value of x=[0,pi/2], a unique value exists in set B.

c) It is not surjective, as |x| will give only +ve values and we won\'t get any value in set A for -ve values in set B.
  It is not injective as every value in set B will have two matching members in set A, e.g: for x=2 and x = -2 both will give 2 as value in set B.

d) It is surjective as sin(1/x) will take all the values from [-1,1] when x goes from (0,1)
It is not injective as for every value from (0,1), sin(1/x) can give more than one value as a result

Injective: (a), (b)
Surjective: (a), (d)