Question

If fi R-R is a
function defined by
(a) = 2x+1 then prove
f is bijection
ove that
3​

Answer 1

Solution :

Given function :

f : R → R , f(x) = 2x + 1

• Whether f(x) is one-one :-

Let f(x1) = f(x2)

=> 2x1 + 1 = 2x2 + 1

=> 2x1 = 2x2

=> x1 = x2

Since f(x1) = f(x2) => x1 = x2 , hence f(x) is one-one function .

• Whether f(x) is onto :-

Let y = f(x)

=> y = 2x + 1

=> 2x = y - 1

=> x = (y - 1)/2

Since the domain of the function is R , thus for x to be real y can be any real number .

=> Range (f) = R

=> Range (f) = Co-domain (f)

Since the range and the Co-domain of the given function are equal , hence f(x) is onto function .

Since the given function f(x) is one-one and as well as onto , thus f(x) is bijective .

Hence proved .