Let f : N → R : f(x) = 4x2 + 12x + 15. Show that f: N → range (f) is invertible. 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) = 4x2 + 12x + 15 So f(x1) = f(x2) (42 + 12 +1 5) = (42 + 12 +1 5), on solving we get x1=x2
So f(x1) = f(x2) x1= x2, f(x) is one-one
Given co-domain of f(x) is Range(f).
Let y = f(x) = 4x2 + 12x + 15, So x = [Range of f(x) = Domain of y]
So Domain of y = Range of f(x) = [6, ∞]
Hence, Range of f(x) = co-domain of f(x) = [6, ∞]
So, f(x) is onto function
As it is bijective function. So it is invertible
Invers of f(x) is f-1(y) =.