question find the equation of all horizontal asymptotes of the following function. f(x)=-2·(8/5)^x + 4

Explanation:

Step1: Recall exponential - function behavior

For an exponential function of the form $y = a\cdot b^{x}+c$, where $b> 0,b\neq1$. When $|b|>1$, as $x\to-\infty$, $b^{x}\to0$. Here $a=-2$ and $b = \frac{8}{5}>1$.

Step2: Find the limit as $x\to-\infty$

We find $\lim_{x\to-\infty}f(x)=\lim_{x\to-\infty}\left(-2\cdot\left(\frac{8}{5}\right)^{x}+4\right)$. Since $\lim_{x\to-\infty}\left(\frac{8}{5}\right)^{x}=0$, then $\lim_{x\to-\infty}\left(-2\cdot\left(\frac{8}{5}\right)^{x}+4\right)= - 2\times0 + 4=4$.

Step3: Find the limit as $x\to+\infty$

As $x\to+\infty$, $\left(\frac{8}{5}\right)^{x}\to+\infty$. So, $\lim_{x\to+\infty}\left(-2\cdot\left(\frac{8}{5}\right)^{x}+4\right)=-\infty$.

Answer:

$y = 4$