Misc. 4 - Chapter 1 Class 12 Relation and Functions

Transcript

Misc. 4 Show that function f: R {x R: 1 < x < 1} defined by f(x) = x 1 + , x R is one-one and onto function. f: R {x R: 1 < x < 1} f(x) = x 1 + We know = , 0 , <0 So, = 1 + , 0 & 1 , <0 Checking one-one Hence, if f(x1) = f(x2) , then x1 = x2 f is one-one = 1 + , 0 & 1 , <0 Checking onto Thus, x = 1 , for x 0 & x = 1 + , for x < 0 Here, y {x R: 1 < x < 1} i.e. Value of y is from 1 to 1 , 1 < y < 1 If y = 1, x = 1 will be not defined, But here 1 < y < 1 So, x is defined for all values of y. & x R f is onto Hence, f is one-one and onto.