Theory question 109
Functions(a) The functions f: x → x2 + 1 and g: x → 5 − 3x are defined on the set of real numbers, R. (i) State the domain of f−1, the inverse of f. (ii) Find g−1(2).
(b) Evaluate ∫02 (x + 3)/√(x2 + 6x + 9) dx
Show the model answer
Model answer
(a)(i) f⁻¹(x) = √(x − 1) (taking x ≥ 0 for f), so the domain of f⁻¹ is {x : x ≥ 1} (the range of f).
(ii) g⁻¹(x) = (5 − x)/3, so g⁻¹(2) = 1.
(b) √(x² + 6x + 9) = x + 3 for 0 ≤ x ≤ 2, so the integrand is 1 and the integral is 2.
Explanation
Notice x² + 6x + 9 = (x + 3)².