Specify the term of a function $j$ defined on $\mathbb { R }$ and invertible that satisfies the following condition: The graph of $j$ and the graph of the inverse function of $j$ have no common point.

Given is the function $f : x \mapsto 2 - \ln ( x - 1 )$ with maximal domain $D _ { f }$. The graph of $f$ is denoted by $G _ { f }$.

\textbf{(1a)} [3 marks] Show that $\left. D _ { f } = \right] 1 ; + \infty [$ and specify the behavior of $f$ at the boundaries of the domain.

\textbf{(1b)} [2 marks] Calculate the zero of $f$.

\textbf{(1c)} [5 marks] Describe how $G _ { f }$ is obtained step by step from the graph of the function $x \mapsto \ln x$ defined in $\mathbb { R } ^ { + }$. Use this to explain the monotonicity behavior of $G _ { f }$.

\textbf{(1d)} [4 marks] Show that $F : x \mapsto 3 x - ( x - 1 ) \cdot \ln ( x - 1 )$ with domain $\left. D _ { F } = \right] 1 ; + \infty [$ is an antiderivative of $f$, and determine the term of the antiderivative of $f$ that has a zero at $x = 2$.

Figure 1 shows an obstacle element in a skate park.

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  \textit{[Figure]}
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\caption{Fig. 1}
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The ramp of the symmetric obstacle element transitions into a horizontally running plateau, which is followed by the descent. The front and rear side surfaces run perpendicular to the horizontal ground. To describe the front side surface mathematically, a Cartesian coordinate system is chosen such that the x-axis represents the lower boundary and the y-axis represents the axis of symmetry of the surface in question. In the model, the plateau extends in the range $- 2 \leq x \leq 2$. The profile line of the descent is described for $2 \leq x \leq 8$ by the graph of the function $f$ investigated in Task 1 (see Figure 2). Here, one unit of length in the coordinate system corresponds to one meter in reality.\\
\textit{[Figure]}

\textbf{(2a)} [2 marks] Explain the meaning of the function value $f ( 2 )$ in the context of the problem and specify the term of the function $q$ whose graph $G _ { q }$ describes the profile line of the ramp in the model for $- 8 \leq x \leq - 2$.

\textbf{(2b)} [5 marks] Calculate the point $x _ { m }$ in the interval [ $2 ; 8$ ] where the local rate of change of $f$ equals the average rate of change over this interval.

\textbf{(2c)} [3 marks] The value $x _ { m }$ determined by calculation in Task 2b could alternatively be determined approximately without calculation using Figure 2. Explain how you would proceed.

\textbf{(2d)} [2 marks] Based on the model, calculate the size of the angle $\alpha$ that the plateau and the roadway enclose at the edge to the descent (see Figure 2).

\textbf{(2e)} [3 marks] The front side surface of the obstacle element is used as advertising space in partial areas of the ramp and descent (see Figure 1). In the model, these are two surface pieces, namely the area between $G _ { f }$ and the x-axis in the range $2 \leq x \leq 6$ and the corresponding symmetric area in the second quadrant. Using the antiderivative $F$ specified in Task 1d, calculate how many square meters are available as advertising space.

Consider the family of functions $g _ { k } : x \mapsto k x ^ { 3 } + 3 \cdot ( k + 1 ) x ^ { 2 } + 9 x$ defined on $\mathbb { R }$ with $k \in \mathbb { R } \backslash \{ 0 \}$ and the corresponding graphs $G _ { k }$. For each $k$, the graph $G _ { k }$ has exactly one inflection point $W _ { k }$.

\textbf{(3a)} [2 marks] Specify the behavior of $g _ { k }$ at the boundaries of the domain in dependence on $k$.

\textbf{(3b)} [3 marks] Determine the x-coordinate of $W _ { k }$ in dependence on $k$.\\
(for verification: $x = - \frac { 1 } { k } - 1$ )

\textbf{(3c)} [4 marks] Determine the value of $k$ such that the corresponding inflection point $W _ { k }$ lies on the y-axis. Show that in this case the point $W _ { k }$ lies at the origin and the inflection tangent, i.e., the tangent to $G _ { k }$ at the point $W _ { k }$, has slope 9.

\textbf{(3d)} [2 marks] For the value of $k$ determined in Task 3c, Figure 3 shows the corresponding graph with its inflection tangent. In this coordinate system, the two axes have different scales.\\
Determine the missing numerical values at the tick marks on the y-axis using an appropriate slope triangle on the inflection tangent and enter the numerical values in Figure 3.\\
\textit{[Figure]}