Question

which statement is true?
a. the function ( f(x) ) has a horizontal asymptote at ( y = -3 ), and the function ( g(x) ) has a horizontal asymptote at ( y = -2 ).
b. the function ( f(x) ) does not have a horizontal asymptote, and the function ( g(x) ) has a horizontal asymptote at ( y = -2 ).
c. the function ( f(x) ) has a horizontal asymptote at ( y = -3 ), and the function ( g(x) ) does not have a horizontal asymptote.
d. the function ( f(x) ) has a horizontal asymptote at ( y = -2 ), and the function ( g(x) ) has a horizontal asymptote at ( y = -1 )

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 \(\lim_{x
ightarrow-\infty}g(x)=L\), 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 \(n=m\), \(\lim_{x
ightarrow\pm\infty}g(x)=\frac{a_n}{b_m}\), if \(n\lt m\), \(\lim_{x
ightarrow\pm\infty}g(x) = 0\).
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, \(\lim_{x
ightarrow-\infty}g(x)=-2\) (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 \(y = - 2\)