The function $\text{g}$ is defined by $\text{g} : x \to \sin x + x$, for $x \in \mathbb{R}$, $-\pi \le x \le \pi$.

1. Find the first derivative of $\text{g}(x)$.

$\text{g}'(x) = \frac{\text{d}}{\text{d}x}(\sin x + x) = \cos x + 1$

2. Show that the derivative is strictly positive on the given open interval.

For $-\pi < x < \pi$, the range of $\cos x$ is $-1 < \cos x \le 1$.
Since $\cos x > -1$, it follows that $\cos x + 1 > 0$.
Hence, $\text{g}'(x) > 0$ for $-\pi < x < \pi$.

3. Deduce that the function has an inverse.

Since $\text{g}'(x) > 0$ for the interior of its domain, $\text{g}$ is a strictly increasing function.
A strictly increasing function is a one-to-one function (it passes the horizontal line test).
Therefore, the inverse function $\text{g}^{-1}$ exists.

1. Identify key coordinates, relative position to $y=x$, and endpoint tangents of $y = \text{g}(x)$.

$\text{g}(-\pi) = -\pi, \quad \text{g}(0) = 0, \quad \text{g}(\pi) = \pi$
Since $\text{g}'(\pm\pi) = \cos(\pm\pi) + 1 = 0$, the graph of $\text{g}$ has horizontal tangents at the endpoints $(-\pi, -\pi)$ and $(\pi, \pi)$.
Furthermore, for $0 < x < \pi$, $\sin x > 0 \implies \text{g}(x) > x$, so the curve lies above the line $y=x$. Conversely, for $-\pi < x < 0$, $\sin x < 0 \implies \text{g}(x) < x$, meaning the curve lies below $y=x$.

2. Deduce the corresponding properties for $y = \text{g}^{-1}(x)$ using the reflection property of inverse functions.

The graph of $y = \text{g}^{-1}(x)$ is the reflection of $y = \text{g}(x)$ in the line $y = x$.
Therefore, it will share the intersection points $(-\pi, -\pi)$, $(0,0)$, and $(\pi, \pi)$.
The horizontal tangents of $\text{g}(x)$ become vertical tangents for $\text{g}^{-1}(x)$ at the endpoints.
For $0 < x < \pi$, $\text{g}^{-1}(x)$ lies below $y=x$, and for $-\pi < x < 0$, it lies above $y=x$.

3. Produce the combined sketch.

(Student draws the axes, sketching the line $y=x$ as a guide, and carefully draws $y = \text{g}(x)$ and $y = \text{g}^{-1}(x)$ exhibiting the correct intersection points, relative positions, and tangent behaviors at the extremities.)