Question

Let f: R → R defined by Is f(x) = (e^x - e^-x)/2 invertible? If so, find its inverse.

Answer 1

To check for invertibility of f(x):

(i)  For one-to-one function: Let x₁x₂ ∈ R and x₁ < x₂. Then

e^x < e^x₂ (As e > 1)   ....(1)

Also,

x₁ < x₂ ⇒ −x₂ < −x_z

⇒ e^-x₂  < e^-x₁  .......(2)

On adding Eqs. (1) and (2), we get

where f is an increasing function. Hence, f(x) is a one-to-one function.

(ii) For onto function: Since x → ∞, f(x) → ∞. Similarly, at x → −∞, f(x) → −∞. That is,

−∞ < f(x) < ∞, x ∈ (−∞, ∞)

Hence, the range of f is the same as the set R. Therefore, f(x) is an onto function. Since f(x) is both one-one and onto functions, f(x) is invertible.

(iii) To find f ⁻¹: We have