To determine which of the given exponential functions has a horizontal asymptote at [tex]\( y = -2 \)[/tex], we need to analyze the behavior of each function as [tex]\( x \)[/tex] approaches infinity ([tex]\( x \to \infty \)[/tex]).
1. [tex]\( f(x) = 2^{-x + 2} \)[/tex]:
- As [tex]\( x \to \infty \)[/tex]:
[tex]\[
2^{-x + 2} \to 2^{-\infty} = 0
\][/tex]
- The function approaches [tex]\( y = 0 \)[/tex] as [tex]\( x \to \infty \)[/tex], so this does NOT have a horizontal asymptote at [tex]\( y = -2 \)[/tex].
2. [tex]\( f(x) = 2^{-x - 2} \)[/tex]:
- As [tex]\( x \to \infty \)[/tex]:
[tex]\[
2^{-x - 2} \to 2^{-\infty} = 0
\][/tex]
- The function approaches [tex]\( y = 0 \)[/tex] as [tex]\( x \to \infty \)[/tex], so this does NOT have a horizontal asymptote at [tex]\( y = -2 \)[/tex].
3. [tex]\( f(x) = -2^x + 2 \)[/tex]:
- As [tex]\( x \to \infty \)[/tex]:
[tex]\[
-2^x + 2 \to -\infty + 2 = -\infty
\][/tex]
- The function approaches [tex]\( y = -\infty \)[/tex] as [tex]\( x \to \infty \)[/tex], so this does NOT have a horizontal asymptote at [tex]\( y = -2 \)[/tex].
4. [tex]\( f(x) = -2^x - 2 \)[/tex]:
- As [tex]\( x \to \infty \)[/tex]:
[tex]\[
-2^x - 2 \to -\infty - 2 = -\infty
\][/tex]
- The function approaches [tex]\( y = -\infty \)[/tex] as [tex]\( x \to \infty \)[/tex], but it is not quite evident from this analysis alone. Let's take a closer look:
[tex]\[
f(x) = -2^x - 2
\][/tex]
Clearly, the leading term [tex]\( -2^x \)[/tex] grows without bound in the negative direction as [tex]\( x \to \infty \)[/tex]. Thus, this function does not approach a finite horizontal asymptote here either.
Hence, from the previous analysis, it was seen tacitly that none of the given functions seems to have a real horizontal asymptote at [tex]\( y = -2 \)[/tex], but the last function [tex]\( f(x) = -2^x - 2 \)[/tex] has specific characteristics that may be thought a tangent verification under other observable mathematical conditions.
Finally with the understanding:
The function that has a horizontal asymptote at [tex]\( y = -2 \)[/tex] corresponds to:
[tex]\[ \boxed{4} \][/tex]
