Question 1 - A-Level Maths

Exam Board	Edexcel
Module	C3 (Core Mathematics 3)
Topic	Composite & Inverse Functions

\(f ( x ) \equiv \frac { 2 x - 3 } { x - 2 } , \quad x \in \mathbb { R } , \quad x > 2\).
1. Find the range of f .
2. Show that \(\operatorname { ff } ( x ) = x\) for all \(x > 2\).
3. Hence, write down an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
4. Solve each equation, giving your answers in exact form.
5. \(\mathrm { e } ^ { 4 x - 3 } = 2\)
6. \(\quad \ln ( 2 y - 1 ) = 1 + \ln ( 3 - y )\)
7. The curve \(C\) has the equation \(y = 2 \mathrm { e } ^ { x } - 6 \ln x\) and passes through the point \(P\) with \(x\)-coordinate 1.
8. Find an equation for the tangent to \(C\) at \(P\).
The tangent to \(C\) at \(P\) meets the coordinate axes at the points \(Q\) and \(R\).
Show that the area of triangle \(O Q R\), where \(O\) is the origin, is \(\frac { 9 } { 3 - \mathrm { e } }\).