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(section:uge2S)=
# Extra (Optional) Exercises -- Long Day

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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.

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## Extra Exercise 1: A Bijection

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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}
:class: dropdown
To get an idea of the behaviour of the function, try calculating $f(1)$, $f(2)$, $f(3)$, $f(4)$ and $f(5)$.
```

```{admonition} Answer
:class: dropdown
Yes.
```

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## Extra Exercise 2: Hyperbolic Functions

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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 $\mathbb R$. 

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### Question a

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


```{hint}
:class: dropdown
Are the functions monotone?
```

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### Question b

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

```{hint}
:class: dropdown
Multiply your equation through with $\mathrm e^x$ and solve the quadratic equation that appears.
```

```{hint}
:class: dropdown
Why can we ignore one of the solutions?
```