O-LevelAdditional MathematicsFunctions4037/21 · May/June 2020 · Q1110 Marks
The function f is defined by f(x) = ln(2x+1) for x ≥ 0.
- (a)Sketch the graph of y = f(x) and hence sketch the graph of y = f⁻¹(x) on the axes below. [Figure 11.1][3]
- (b)(i)The function g is defined by g(x) = (x-4)² + 1 for x ≤ 4. Find an expression for g⁻¹(x) and state its domain and range.[4]
- (b)(ii)Find and simplify an expression for fg(x).[2]
- (b)(iii)Explain why the function gf does not exist.[1]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.
- [B3] B1 for correct shape of f or f⁻¹
- B1 for symmetry
- B1 for drawn over correct domain
- Maximum of 2 marks if not fully correct
- [M1] [±]√(x-1)=y-4 soi
- [A1] g⁻¹(x)=4-√(x-1)
- [B1] [Range] g⁻¹ < 4
- [B1] [Domain] x ≥ 1
- [M1] ln(2[(x-4)² + 1] + 1)
- [A1] ln(2x² - 16x +35)
- [B1] Valid explanation, e.g. some of the values in the range of f are outside the domain of g
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