1) Let's graph or plot that rational function
[tex]f(x)=\frac{1}{4+x}[/tex]
Note that the vertical asymptote is x=-4. Since the asymptote is x=-4, then the domain is given by x < -4 or x >-4, in interval notation, we can write
D = (-∞, -4) U (-4, ∞). The Range, f(x) or y <0 or y> 0 R= (-∞, 0) U (0, ∞).
As we can notice the graph does not touch the x-axis, so the horizontal asymptote is y=0
2) Now let's compare it to its parent function in blue
What we have here is a Translation to the left 2 units.
3) So in short the answers are:
Translation to the left 2 units:
[tex]T_{-2}[/tex]
vertical asymptote is x=-4
horizontal asymptote is y=0
D = (-∞, -4) U (-4, ∞)
R= (-∞, 0) U (0, ∞).
