Recovering a function from a composite
Recovering a function from a composite
7 marks
TGM Original
It is given that \(\mathrm{f}(x)=3x-2\) and \(\mathrm{fg}(x)=3\mathrm{e}^{2x}+4\), for \(x\in\mathbb{R}\).
- Find \(\mathrm{f}^{-1}(x)\).[1]
- Hence find \(\mathrm{g}(x)\).[2]
A different pair of functions satisfies \(\mathrm{q}(x)=1+\mathrm{e}^{x}\) and \(\mathrm{pq}(x)=\mathrm{e}^{2x}+4\mathrm{e}^{x}+1\), for \(x\in\mathbb{R}\).
- By writing \(u=\mathrm{q}(x)\), find \(\mathrm{p}(u)\) in terms of \(u\). Hence state \(\mathrm{p}(x)\).[3]
- State the set of values of \(x\) for which this expression for \(\mathrm{p}(x)\) is determined by the given information, and explain why.[1]
Solution:
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Answer:(a) \(\mathrm{f}^{-1}(x)=\dfrac{x+2}{3}\) (b) \(\mathrm{g}(x)=\mathrm{e}^{2x}+2\) (c) \(\mathrm{p}(u)=u^2+2u-2\), \(\mathrm{p}(x)=x^2+2x-2\) (d) \(x>1\), the range of \(\mathrm{q}\)
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