A function is defined by \(\mathrm f(x)=\dfrac{x^2+20x+101}{x+10}\), where \(x\in\mathbb R\), \(x\neq-10\).
Express \(\mathrm f(x)\) in the form \(ax+b+\dfrac c{x+10}\), where \(a,b,c\) are constants to be determined.[3]
Write down the equations of the asymptotes of the graph \(y=\mathrm f(x)\).[2]
Prove that the graph of \(y=\mathrm f(x)\) does not cut the \(x\)-axis.[3]
Sketch the graph of \(y=\mathrm f(x)\), labelling clearly the equations of the asymptotes, the coordinates of the turning points and the intersections with the axes.[4]
Solve the inequality \(\mathrm f(2x)\geq x\).[3]
Hence solve \(\mathrm f(2\ln x)\geq\ln x\). Give your answers to 6 decimal places.[3]