We are asked to determine the function that has a vertical asymptote at x = -4 and a horizontal asymptote at y = 2. To do that we need to have into account that the vertical asymptotes are values of "x" that the function can't take as input.
Since we are given rational functions, this means that the values of "x" that the function can't take as input are the ones where the denominator is equal to zero, therefore, the denominator of the function must be of the form:
[tex]\begin{gathered} x=4 \\ x-4=0 \end{gathered}[/tex]
The denominator must be "x - 4". That leaves us with options A or C.
Now, the horizontal asymptote. We will take option A and we will add the fraction and the whole number:
[tex]f(x)=\frac{1}{x+4}+2[/tex]
Adding the fraction:
[tex]f(x)=\frac{1+2x+8}{x+4}[/tex]
We notice that the numerator and denominator are polynomials of the same degree, 1. When we have a fraction where the numerator and denominator are polynomials of the same degree then the horizontal asymptote is the leading coefficient of the numerator over the leading coefficient of the denominator.
Therefore, the horizontal asymptote is:
[tex]y=\frac{2}{1}=2[/tex]
Therefore, the right option is A.
