Question

I need help with this question in the picture below, I need to find the transformation , vertical asymptote, horizontal asymptote, domain and range

Answer 1

1) Let's graph or plot that rational function

`f(x)=(1)/(4+x)`

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:

`T_(-2)`

vertical asymptote is x=-4

horizontal asymptote is y=0

D = (-∞, -4) U (-4, ∞)

R= (-∞, 0) U (0, ∞).