The graph in the figure, representing the continuous function $y = f(x)$, is the union of the parabolic arc $\Gamma_{1}$, the circular arc $\Gamma_{2}$ and the hyperbolic arc $\Gamma_{3}$.
a) Write an analytical expression of the function $f$ defined piecewise on the interval $[-2; 2]$, using the equations:
$$y = a(x + 2)^{2} \quad x^{2} + y^{2} + b = 0 \quad x^{2} - y^{2} + c = 0$$
and identify the appropriate values for the real parameters $a, b, c$.
Study the differentiability of the function $f$ and write the equations of any tangent lines at the points with abscissa
$$x = -2 \quad x = 0 \quad x = 1 \quad x = 2$$
b) Starting from the graph of the function $f$, deduce that of its derivative $f^{\prime}$ and identify the intervals of concavity and convexity of $F(x) = \int_{-2}^{x} f(t) dt$.
c) Consider the function $y = \frac{1}{4}(x + 2)^{2}$, defined on the interval $[-2; 0]$, of which $\Gamma_{1}$ is the representative graph. Explain why it is invertible and write the analytical expression of its inverse function $h$. Study the differentiability of $h$ and sketch its graph.
d) Let $S$ be the bounded region in the second quadrant, between the graph $\Gamma_{1}$ and the coordinate axes. Determine the value of the real parameter $k$ so that the line with equation $x = k$ divides $S$ into two equivalent regions.