Question

§4.5: use arrow notation to describe the end behavior of the function.

1. $g(x) = -\frac{7x^2 + 5x}{11x^2 - 3x + 16}$

Explanation:

Step1: Identify Degree and Leading Coefficients

For a rational function \( g(x)=\frac{a_nx^n + \dots}{b_mx^m+\dots} \), the end - behavior is determined by the degrees of the numerator (\(n\)) and the denominator (\(m\)) and their leading coefficients (\(a_n\) and \(b_m\)).

For the function \(g(x)=-\frac{7x^{2}+5x}{11x^{2}-3x + 16}\), the degree of the numerator \(n = 2\) (the highest power of \(x\) in the numerator) and the degree of the denominator \(m=2\) (the highest power of \(x\) in the denominator). The leading coefficient of the numerator \(a_n=- 7\) (the coefficient of \(x^{2}\) in the numerator, considering the negative sign in front of the fraction) and the leading coefficient of the denominator \(b_m = 11\) (the coefficient of \(x^{2}\) in the denominator).

Step2: Analyze End - Behavior for \(n = m\)

When the degree of the numerator is equal to the degree of the denominator (\(n=m\)), the end - behavior of the rational function \(y = g(x)\) is given by the ratio of the leading coefficients. That is, as \(x
ightarrow\pm\infty\), \(g(x)
ightarrow\frac{a_n}{b_m}\).

For our function, \(\frac{a_n}{b_m}=\frac{- 7}{11}\).

So, as \(x
ightarrow\infty\), we look at the limit of \(g(x)\) as \(x\) approaches positive infinity. Since \(n = m\), \(\lim_{x
ightarrow\infty}g(x)=\frac{-7}{11}\). In arrow notation, this is \(x
ightarrow\infty\Rightarrow g(x)
ightarrow-\frac{7}{11}\).

As \(x
ightarrow-\infty\), we look at the limit of \(g(x)\) as \(x\) approaches negative infinity. Again, since \(n = m\), \(\lim_{x
ightarrow-\infty}g(x)=\frac{-7}{11}\). In arrow notation, this is \(x
ightarrow-\infty\Rightarrow g(x)
ightarrow-\frac{7}{11}\).

Answer:

As \(x
ightarrow\infty\), \(g(x)
ightarrow-\frac{7}{11}\) and as \(x
ightarrow-\infty\), \(g(x)
ightarrow-\frac{7}{11}\) (or in arrow notation: \(x
ightarrow\infty\Rightarrow g(x)
ightarrow-\frac{7}{11}\), \(x
ightarrow-\infty\Rightarrow g(x)
ightarrow-\frac{7}{11}\))