Answer
(a)
1st step (black graph): The parent function: $\dfrac{1}{x}$
2nd step (green graph): Flipped over the x-axis. $-\dfrac{1}{x}$
3rd step (red graph): Stretched by a factor of 2. $-\dfrac{2}{x}$
Final step (blue graph): Translated by 1 unit to the left. $-\dfrac{2}{x+1}$
(b) The domain is $(-\infty,-1)\cap(-1,\infty)$
The range is $(-\infty,0)\cap(0,\infty)$
(c) The only horizontal asymptote is y=0.
The only vertical asymptote is x=-1.
There are no oblique asymptotes.
Work Step by Step
The domain is a horizontal span from the function's smallest value of x to the function's largest value of x. If there is a discontinuity, the domain must show where the discontinuity happens. For example, if there is a vertical asymptote on x=3, the domain would be $(-\infty,3)\cap(3,\infty)$
The range is a vertical span from the function's smallest value of f(x) to the function's largest value of f(x). If there is a discontinuity, the range must show where the discontinuity happens. For example, if there is a horizontal asymptote on y=-4, the domain would be $(-\infty,-4)\cap(-4,\infty)$
The x-intercepts are all points of a graph when f(x)=0 while the y-intercepts are all points of a graph when x=0.
Horizontal asymptotes are horizontal lines that approach a graph but never intersect it.
Vertical asymptotes are vertical lines that approach a graph but never intersect it.
Oblique asymptotes are diagonal lines that approach a graph and may intersect it.