\textbf{(1)} [4 marks] Given is the function $f$ defined on $\mathbb { R }$ with $f ( x ) = e ^ { 2 x + 1 }$. Show that $f$ is invertible and determine a term for the inverse function of $f$.

\textbf{(2a)} [3 marks] Given is the function $g : x \mapsto \left( x ^ { 2 } - 9 x \right) \cdot \sqrt { 2 - x }$ with maximal domain $D _ { g }$. State $D _ { g }$ and all zeros of $g$.

\textbf{(2b)} [3 marks] Given is the function $h : x \mapsto \ln \left( \frac { 1 } { x ^ { 2 } + 1 } \right)$ defined on $\mathbb { R }$. Justify that the range of $h$ is the interval ] $- \infty ; 0 ]$.

Consider the function $f$ defined on $\mathbb { R } ^ { + }$ with $f ( x ) = \frac { 1 } { \sqrt { x ^ { 3 } } }$.\\

\textbf{(3a)} [2 marks] Show that the function $F$ defined on $\mathbb { R } ^ { + }$ with $F ( x ) = - \frac { 2 } { \sqrt { x } }$ is an antiderivative of $f$.

\textbf{(3b)} [3 marks] The graph of $f$ encloses an area with the x-axis and the lines with equations $x = 1$ and $x = b$ with $b > 1$. Determine the value of $b$ for which this area has content 1.

Given are the function $f$ defined on $\mathbb { R }$ with $f ( x ) = \frac { 1 } { 8 } x ^ { 3 }$ and the points $Q _ { a } ( a \mid f ( a ) )$ for $a \in \mathbb { R }$. The figure shows the graph of $f$ as well as the points $P ( 0 \mid 2 )$ and $Q _ { 2 }$.\\
\textit{[Figure]}

\textbf{(4a)} [2 marks] Calculate for $a \neq 0$ the slope $m _ { a }$ of the line through the points $P$ and $Q _ { a }$ as a function of $a$.\\
(for verification: $m _ { a } = \frac { a ^ { 3 } - 16 } { 8 a }$ )

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The tangent to the graph of $f$ at the point $Q _ { a }$ is denoted by $t _ { a }$. Determine computationally the value of $a \in \mathbb { R }$ for which $t _ { a }$ passes through $P$.

Given is the function $f : x \mapsto \frac { 6 x } { x ^ { 2 } - 4 }$ defined on $\mathbb { R } \backslash \{ - 2 ; 2 \}$. The graph of $f$ is denoted by $G _ { f }$ and is symmetric with respect to the origin.\\