If f : R → R be defined as f x = e 2 x - e - 2 x 2 , then

Step-by-step solution

Let us check for invertibility of f x A one-one: we have, f x = e 2 x - e - 2 x 2 ⇒ f ′ x = e 4 x + 1 e 2 x , which is strictly increasing as e 4 x > 0 for all x . Thus, f is one-one B Onto: Let y = f x ⇒ d y d x = e 2 x + e - 2 x , where y is strictly monotonic Hence, the range of f x = f - ∞ , f ∞ ⇒ range of f x = - ∞ , ∞ So, the range of f x = co-domain Hence, f x is one-one and onto C To find f - 1 : y = e 4 x - 1 2 e 2 x ⇒ e 4 x - 2 e 2 x y - 1 = 0 &#86