O-LevelAdditional MathematicsFunctions4037/22 · May/June 2019 · Q129 Marks
(a) The functions f and g are defined by `f(x) = 5x-2` for `x > 1`, `g(x) = 4x²-9` for `x > 0`. (b) The function h is defined by `h(x) = √(x²-1)` for `x ≤ -1`.
- (a)(i)State the range of g.[1]
- (a)(ii)Find the domain of gf.[1]
- (a)(iii)Showing all your working, find the exact solutions of `gf(x) = 4`.[3]
- (b)(i)State the geometrical relationship between the graphs of `y = h(x)` and `y = h⁻¹(x)`.[1]
- (b)(ii)Find an expression for `h⁻¹(x)`.[3]
Mark scheme
Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.
- [B1] (a)(i) g > -9
- [B1] (a)(ii) x > 1
- [B1] (a)(iii) [gf(x) =] 4(5x-2)²-9
- [M1] 100x²-80x-38=0 or (5x-2)² = 45+9/4
- [A1] (4+3√6)/10 oe only (leading to (4+3√6)/10 oe only or 1/5 (2 + √(54/4)) or better only)
- [B1] (b)(i) (They are) reflections (of each other) in (the line) y = x oe
- [M1] (b)(ii) x² = y² + 1 or y² = x² + 1
- [A1] x=[±]√(y²+1) or y=[±]√(x²+1)
- [A1] -√(x²+1) nfww
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