Determine the horizontal asymptote of $$ f(x)=2^{x-2} $$ Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is $$ \square $$ and the bottom asymptote is $$ \square $$ (Type equations.) C. The function has no horizontal asymptotes.

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UpStudy AI · Accepted Answer

The function is given by

$$
f(x) = 2^{x-2}.
$$

We can rewrite it as

$$
f(x) = \frac{2^x}{2^2} = \frac{2^x}{4}.
$$

To find horizontal asymptotes, we examine the behavior of $$ f(x) $$ as $$ x $$ approaches both $$ \infty $$ and $$-\infty$$.

1. As $$ x \to \infty $$:
   $$
   f(x) = \frac{2^x}{4} \to \infty,
   $$
   so there is no horizontal asymptote in this direction.

2. As $$ x \to -\infty $$:
   $$
   f(x) = \frac{2^x}{4} \to \frac{0}{4} = 0,
   $$
   because $$2^x \to 0$$ as $$x \to -\infty$$.

Thus, the function has one horizontal asymptote, which is

$$
y = 0.
$$

The correct choice is A with the equation $$y = 0$$.
The function $$ f(x) = 2^{x-2} $$ has one horizontal asymptote at $$ y = 0 $$.

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