To prove one-one & onto (injective, surjective, bijective)

Chapter 1 Class 12 Relation and Functions
Concept wise

### Transcript

Ex 1.2, 1 Show that the function f: R* β R* defined by f(x) = 1/x is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R*? Solving for f: R* β R* f(x) = 1/x Checking one-one f (x1) = 1/π₯1 f (x2) = 1/π₯2 Putting f (x1) = f (x2) 1/x1 = 1/x2 x2 = x1 Hence, if f(x1) = f(x2) , x1 = x2 β΄ f is one-one Check onto f: R* β R* f(x) = 1/π₯ Let y = f(x) , such that y β R* y = 1/π₯ x = 1/π¦ Since y not equal to 0, x is possible Thus we can say that if y β R β {0} , then x β R β {0} also Thus, For every y β R* , there exists x β R* such that f(x) = y Hence, f is onto Show that the function f: R* β R* defined by f(x) = 1/x is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R*? Now, domain R* is replaced by N , codomain remains R* Hence f : N β R* f(x) = 1/x Checking one-one f (x1) = 1/x1 f (x2) = 1/x2 f (x1) = f (x2) 1/x1 = 1/x2 x2 = x1 Hence, if f(x1) = f(x2) , x1 = x2 β΄ f is one-one Check onto f: N β R* f(x) = 1/x Let y = f(x) , , such that y β R* y = 1/π₯ x = 1/π¦Since y is real number except 0, x cannot always be a natural number Example For y = 2 x = 1/2 So, x is not a natural number Hence, f is not onto

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.