Show that the function f : R → R : f(x) = 2x + 3 is invertible and find f-1.
To Show: that f is invertible
To Find: Inverse of f
[NOTE: Any functions is invertible if and only if it is bijective functions (i.e. one-one and onto)]
one-one function: A function f : A B is said to be a one-one function or injective mapping if different elements of A have different images in B. Thus for x1, x2 A & f(x1), f(x2) B, f(x1) = f(x2) x1= x2 or x1 x2f(x1) f(x2)
onto function: If range = co-domain then f(x) is onto functions.
So, We need to prove that the given function is one-one and onto.
Let x1, x2 R and f(x) = 2x+3.So f(x1) = f(x2) 2x1+3 = 2x2+3 x1=x2
So f(x1) = f(x2) x1= x2, f(x) is one-one
Given co-domain of f(x) is R.
Let y = f(x) = 2x+3 , So x = [Range of f(x) = Domain of y]
So Domain of y is R(real no.) = Range of f(x)
Hence, Range of f(x) = co-domain of f(x) = R
So, f(x) is onto function
As it is bijective function. So it is invertible
Invers of f(x) is f-1(y) =