The functions f and g are defined by f(x) = x3 + 1, 0 ≤ x ≤ 3, g(x) = x + 5, x ∈ R where R is the set of real numbers. State the range of g(f(x))

Determine the inverse of g(f(x)).

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .