which statement is true?\na. the function ( f(x) ) has a horizontal asymptote at ( y = -3 ), and the…

Explanation:

Step1: Analyze the horizontal asymptote of (f(x))

From the graph of (f(x)), as (x) approaches (\pm\infty), the function (f(x)) approaches (y = - 3). So, (y=-3) is the horizontal asymptote of (f(x)).

Step2: Analyze the horizontal asymptote of (g(x))

For a function (y = g(x)), if (\lim_{x\rightarrow-\infty}g(x)=L), then (y = L) is a horizontal asymptote. We know that for a rational function (g(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}), if (n=m), (\lim_{x\rightarrow\pm\infty}g(x)=\frac{a_n}{b_m}), if (n\lt m), (\lim_{x\rightarrow\pm\infty}g(x) = 0). Let's assume (g(x)) is a rational function. We can also use the table - values. As (x) gets more and more negative ((x=-4,x = - 3,x=-2,x=-1)), (g(x)) approaches (y=-2). For example, (\lim_{x\rightarrow-\infty}g(x)=-2) (by looking at the trend of (g(x)) values as (x) takes large negative values: (-1.988\approx - 2,-1.963\approx - 2,-1.889\approx - 2,-1.667\approx - 2))

Answer:

A. The function (f(x)) has a horizontal asymptote at (y=-3), and the function (g(x)) has a horizontal asymptote at (y = - 2)