question find the equation of all horizontal asymptotes of the following function. f(x)=-4·(2)^x answer…

Explanation:

Step1: Recall horizontal - asymptote rules for exponential functions

For an exponential function of the form $y = a\cdot b^{x}+k$ ($a\neq0$, $b > 0$, $b\neq1$), we find the horizontal asymptote by considering the limit as $x\to\pm\infty$. The given function is $f(x)=-4\cdot(2)^{x}$, which is in the form $y = a\cdot b^{x}$ with $a=-4$ and $b = 2$.

Step2: Calculate the limit as $x\to+\infty$

We know that $\lim_{x\to+\infty}b^{x}=\infty$ when $b>1$. Here $b = 2>1$, so $\lim_{x\to+\infty}-4\cdot(2)^{x}=-\infty$.

Step3: Calculate the limit as $x\to-\infty$

We know that $\lim_{x\to-\infty}b^{x}=0$ when $b > 1$. So, $\lim_{x\to-\infty}-4\cdot(2)^{x}=-4\times0 = 0$.

Answer:

$y = 0$