Question

A function f:[1, ∞) → [1, ∞) is defined as f(x) = 2^{x(x – 1)}. Find f^–1(x).

Answer 1

Given f(x) = 2^x(x–1).  

⇒ f'(x) = 2^{x(x– 1)} × (2x – 1) × loge 2 > 0 for all x in [1, ∞) 

⇒ f is strictly increasing function. 

⇒ f is one-one function.

Also, R_f = [1, ∞) 

⇒ R_f = [1, ∞) = Co-domain

⇒ f is onto function.

Thus, f is a bijective function.

So its inverse is exists.

Let y = 2^{x(x – 1)}

⇒ y = 2^{x2 – x}