Question

select the correct answer.
function ( r ) is a continuous rational function with a horizontal asymptote at ( y = -8 ).
which statement describes the key features of ( s(x) = r(x + 2) - 1 )?

a. function ( s ) has a point of discontinuity at ( x = -2 ) and a horizontal asymptote at ( y = -9 ).
b. function ( s ) has a point of discontinuity at ( x = -2 ) and a horizontal asymptote at ( y = -1 ).
c. function ( s ) is continuous and has a horizontal asymptote at ( y = -1 ).
d. function ( s ) is continuous and has a horizontal asymptote at ( y = -9 ).

Explanation:

Step1: Analyze continuity of $s(x)$

Since $r(x)$ is continuous everywhere, a horizontal shift $r(x+2)$ and vertical shift $r(x+2)-1$ do not introduce discontinuities. So $s(x)$ is continuous.

Step2: Find horizontal asymptote of $s(x)$

For rational functions, horizontal asymptotes shift vertically with the function. Given $\lim_{x \to \pm\infty} r(x) = -8$, then:
$$\lim_{x \to \pm\infty} s(x) = \lim_{x \to \pm\infty} [r(x+2)-1] = -8 - 1 = -9$$

Answer:

D. Function s is continuous and has a horizontal asymptote at $y = -9$.