O-LevelAdditional MathematicsDifferentiation and integration4037/21 · Oct/Nov 2010 · Q1210 Marks
Answer only one of the following two alternatives. EITHER The functions f and g are defined, for x > 1, by f(x) = (x + 1)² - 4, g(x) = (3x + 5) / (x - 1) Find
- (i)fg(9),[2]
- (ii)expressions for f⁻¹(x) and g⁻¹(x),[4]
- (iii)the value of x for which g(x) = g⁻¹(x).[4]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2010). The mark or mark type is shown in brackets.
- [M1] (i) fg(9) = f(4) evaluated or fg(x) = ( (3x+5)/(x-1) + 1 )² - 4
- [A1] (i) 21
- [M1] (ii) Method for f⁻¹(x)
- [A1] (ii) f⁻¹(x) = √x+4 - 1
- [M1] (ii) Put y = (3x+5)/(x-1) and rearrange
- [A1] (ii) g⁻¹(x) = (x+5)/(x-3)
- [M1] (iii) Rearrange two of (3x+5)/(x-1) = (x+5)/(x-3) to quadratic equation
- [A1] (iii) 2(x² - 4x - 5) = 0
- [M1] (iii) Solve 3 term quadratic
- [A1] (iii) 5 only
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