(type your answer in interval notation.) determine the horizontal asymptote of f(x)=4^(-x)-2. select the…

Explanation:

Step1: Recall the definition of horizontal asymptote

We find the limit as $x\to\pm\infty$. For $y = 4^{-x}-2=\left(\frac{1}{4}\right)^{x}-2$.

Step2: Calculate $\lim_{x\to+\infty}f(x)$

As $x\to+\infty$, we know that for an exponential function $y = a^{x}$ where $0 < a<1$ (here $a=\frac{1}{4}$), $\lim_{x\to+\infty}a^{x}=0$. So $\lim_{x\to+\infty}\left(\frac{1}{4}\right)^{x}-2=0 - 2=-2$.

Step3: Calculate $\lim_{x\to-\infty}f(x)$

As $x\to-\infty$, $\left(\frac{1}{4}\right)^{x}=4^{|x|}\to+\infty$. So the limit as $x\to-\infty$ does not exist as a finite - value.

Answer:

A. The function has one horizontal asymptote, $y = - 2$.