Theory question 111
Functions(a) If f(x) = (x − 3)/(2x − 1), x ≠ ½ and g(x) = (x − 1)/(x + 1), x ≠ −1, find g o f.
(b)(i) Sketch the curve y = 9x − x3. (ii) Calculate the total area bounded by the x-axis and the curve y = 9x − x3.
Show the model answer
Model answer
(a) g(f(x)) = (f − 1)/(f + 1) = [(x − 3) − (2x − 1)]/[(x − 3) + (2x − 1)] = (−x − 2)/(3x − 4) = −(x + 2)/(3x − 4), x ≠ ½, 4/3.
(b)(i) Cuts the x-axis at −3, 0, 3; minimum at x = −√3, maximum at x = √3; falls to the right.
(ii) Area = 2∫₀³ (9x − x³) dx = 2(81/2 − 81/4) = 81/2 = 40.5 square units.
Explanation
Area below the axis is counted as positive.