given $f(x)=\frac{4x - 7}{8x + 8}$, what is the end - behavior of the function?\nas $x\\to-\\infty$…

given $f(x)=\frac{4x - 7}{8x + 8}$, what is the end - behavior of the function?\nas $x\\to-\\infty$, $f(x)\\to0.5$; as $x\\to\\infty$, $f(x)\\to0.5$.\nas $x\\to-\\infty$, $f(x)\\to - 0.5$; as $x\\to\\infty$, $f(x)\\to - 0.5$.\nas $x\\to-\\infty$, $f(x)\\to0.73$; as $x\\to\\infty$, $f(x)\\to0.73$.\nas $x\\to-\\infty$, $f(x)\\to - 0.73$; as $x\\to\\infty$, $f(x)\\to - 0.73$.

given $f(x)=\frac{4x - 7}{8x + 8}$, what is the end - behavior of the function?\nas $x\\to-\\infty$, $f(x)\\to0.5$; as $x\\to\\infty$, $f(x)\\to0.5$.\nas $x\\to-\\infty$, $f(x)\\to - 0.5$; as $x\\to\\infty$, $f(x)\\to - 0.5$.\nas $x\\to-\\infty$, $f(x)\\to0.73$; as $x\\to\\infty$, $f(x)\\to0.73$.\nas $x\\to-\\infty$, $f(x)\\to - 0.73$; as $x\\to\\infty$, $f(x)\\to - 0.73$.

Answer

Explanation:

Step1: Identify the degrees of numerator and denominator

The degree of the numerator $4x - 7$ is 1 and the degree of the denominator $8x + 8$ is 1.

Step2: Use the rule for rational - functions with equal degrees

For a rational function $y=\frac{a_nx^n+\cdots+a_0}{b_nx^n+\cdots + b_0}$ where $n$ is the degree of the numerator and denominator, the horizontal asymptote (which gives the end - behavior) is $y = \frac{a_n}{b_n}$. Here $a_n = 4$ and $b_n=8$.

Step3: Calculate the ratio

$\frac{a_n}{b_n}=\frac{4}{8}=0.5$. So as $x\to-\infty$ and as $x\to\infty$, $f(x)\to0.5$.

Answer:

As $x\to-\infty$, $f(x)\to0.5$; as $x\to\infty$, $f(x)\to0.5$.