O-LevelAdditional MathematicsFunctions4037/22 · May/June 2010 · Q1210 Marks
Answer only one of the following two alternatives. OR The functions f, g and h are defined, for x ∈ R, by f(x) = x² + 1, g(x) = 2x - 5, h(x) = 2ˣ.
- (i)Write down the range of f.[1]
- (ii)Find the value of gf(3).[2]
- (iii)Solve the equation fg(x) = g⁻¹(15).[5]
- (iv)On the same axes, sketch the graph of y = h(x) and the graph of the inverse function y = h⁻¹(x), indicating clearly which graph represents h and which graph represents h⁻¹.[2]
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Mark scheme
Marking points from the official mark scheme (May/June 2010). The mark or mark type is shown in brackets.
- B1 for (f) ≥ 1
- M1 for gf(3)=g(10) or gf(x) = 2(x²+1)-5
- A1 for 15
- B1 for g⁻¹(15) = 10
- M1 for Finds fg(x)
- A1 for (2x-5)² + 1 or 4x² - 20x + 26
- M1 for Solves (2x-5)²+1=10 or 4x² −20x+16=0
- A1 for 1 and 4
- B1 for complete, labelled exponential curve
- B1 for complete logarithmic curve not cutting exponential curve
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