\(\mathrm{f}(x)=\dfrac{x}{x-1}\) for \(-10\leq x\leq10,\ x\ne1\).
The diagram shows the graph of \(y=\mathrm{f}(x)\).
Use the diagram to explain why \(\mathrm{f}\) is a function.[1]
Find \(\mathrm{ff}(x)\), giving your answer in its simplest form.[2]
Using your answer to part (ii) state the relationship between the functions \(\mathrm{f}\) and \(\mathrm{f}^{-1}\).[1]
Explain how the diagram shows the relationship between \(\mathrm{f}\) and \(\mathrm{f}^{-1}\).[1]
A function \(\mathrm{g}\) is defined by \(\mathrm{g}(x)=\dfrac{x}{x-1}\) for \(x\geq2\). Find the range of \(\mathrm{g}\).[1]
A function \(\mathrm{h}\) is defined by \(\mathrm{h}(x)=\dfrac{2x}{3x+1}\) for the largest possible domain. State the domain of \(\mathrm{h}\).[1]
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Answer:(a)(i) Vertical-line test; (ii) \(x\) on the composition domain; (iii) same inverse formula; (iv) visible symmetry about \(y=x\). (b) \(1<\mathrm{g}\leq2\). (c) \(x\in\mathbb R,x\ne-\dfrac13\).