Classify the following functions as injection, surjection or bijection:

f : R → R, defined by f(x) = |x|

TIP: – One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.

So, is One – One function

⇔ a≠b

⇒ f(a)≠f(b) for all

⇔ f(a) = f(b)

⇒ a = b for all

Onto Function: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.

So, is Surjection iff for each , there exists such that f(a) = b

Bijection Function: – A function is said to be a bijection function if it is one – one as well as onto function.

Now, f : R → R, defined by f(x) = |x|

Check for Injectivity:

Let x,y be elements belongs to R i.e such that

Case i

⇒ x = y

⇒ |x| = |y|

Case ii

⇒ – x = y

⇒ | – x| = |y|

⇒ x = |y|

Hence from case i and case ii f is not One – One function

Check for Surjectivity:

Since f attain only positive values, for negative real numbers in R

(co – domain) there is no pre–image in domain R.

Hence, f is not onto function

Thus, Not Bijective also