Extra (Optional) Exercises – Long Day

These exercises are optional extra material for students who are done with the exercises of the day but have the time for and the interest in 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 that are formed from already known functions. The two functions are called hyperbolic sinus and hyperbolic cosinus and they are 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 choose their domains and co-domains to be \(\mathbb R\).

Question a

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

Hint

Are the functions monotone?

Question b

Determine a functional expression for \(\mathrm{sinh}^{-1}\) by isolating \(x\) in the equation \(y=\mathrm{sinh}(x)\).

Hint

Multiply your equation through with \(\mathrm e^x\) and solve the quadratic equation that appears.

Hint

Why can we ignore one of the solutions?