Extra (optional) exercises – Long Day

These exercises are optional extra material for students who have finished the exercises of the day and have the time and interest for more.

Extra exercise 1: A bijection

A function \(f :\Bbb N\rightarrow \Bbb Z\) is defined by

\( x \mapsto \left\{ \begin{array}{ll} \frac{x}{2} & \text{when }x \text{ is even,} \\ -\frac{x-1}{2} & \text{when }x \text{ is odd.} \\ \end{array} \right. \)

Is \(f\) a bijection?

Hint

To get an idea of the behaviour of the function, try calculating \(f(1)\), \(f(2)\), \(f(3)\), \(f(4)\) and \(f(5)\).

Answer

Yes.

Extra exercise 2: Hyperbolic functions

In this exercise we will introduce two new functions built from functions we already know. The two functions are called hyperbolic sine and hyperbolic cosine, defined by:

\[\mathrm{sinh}(x)=\frac{\mathrm e^x-\mathrm e^{-x}}{2}\quad \text{and}\quad \mathrm{cosh}(x)=\frac{\mathrm e^x+\mathrm e^{-x}}{2}.\]

For both functions, we take the domain and codomain to be \(\mathbb R\).

Question a

Show that \(\mathrm{sinh}\) is injective and that \(\mathrm{cosh}\) is not.

Hint

Are the functions monotone?

Question b

In fact, \(\mathrm{sinh}\) is also surjective, so it has an inverse function. Determine an expression for this inverse function \(\mathrm{sinh}^{-1}\) by solving the equation \(y=\mathrm{sinh}(x)\) for \(x\).

Hint

Multiply both sides by \(\mathrm e^x\) and solve the resulting quadratic equation in \(\mathrm e^x\).

Hint

Why can we ignore one of the solutions?