方程为 $y = \dfrac{1}{x+1}$ 的曲线依次经历以下变换： A: 关于 x 轴反射， B: 沿 y 轴方向平移 1 个单位， C: 沿平行于 y 轴的方向伸缩，比例因子为 2。 写出所得曲线的方程。 一曲线依次经历上述变换 A、B 和 C 后，所得曲线的方程为 $y = 2 - |2x+6|$。求变换前该曲线的方程。

1. Apply the sequence of transformations to the curve $y = \frac{1}{x+1}$.

Applying Transformation A (reflection in the $x$-axis):
$y = -\frac{1}{x+1}$

Applying Transformation B (translation of magnitude 1 unit in the $y$-direction):
$y = -\frac{1}{x+1} + 1 = 1 - \frac{1}{x+1}$

Applying Transformation C (scaling parallel to the $y$-axis by a factor of 2):
$y = 2\left(1 - \frac{1}{x+1}\right) = 2 - \frac{2}{x+1}$

2. Find the combined transformation for a general curve $y = \mathrm{f}(x)$.

For a general curve $y = \mathrm{f}(x)$, undergoing the same succession of transformations:
After A: $y = -\mathrm{f}(x)$
After B: $y = -\mathrm{f}(x) + 1$
After C: $y = 2(-\mathrm{f}(x) + 1) = 2 - 2\mathrm{f}(x)$

3. Equate the transformed general function to the given resulting curve and solve for the original curve equation.

We are given that the resulting curve is $y = 2 - |2x+6|$. Equating this to the combined transformation:
$2 - 2\mathrm{f}(x) = 2 - |2x+6|$
$-2\mathrm{f}(x) = -|2x+6|$
$2\mathrm{f}(x) = |2x+6|$

Solving for $\mathrm{f}(x)$ and simplifying the modulus expression:
$\mathrm{f}(x) = \frac{1}{2}|2(x+3)|$
$\mathrm{f}(x) = |x+3|$

The equation of the original curve is $y = |x+3|$.