To analyze the given function [tex]\( f(x) = e^{-x} + 9 \)[/tex], let's proceed step-by-step.
1. Identify the Horizontal Asymptote:
For an exponential function of the form [tex]\( f(x) = e^{-x} + c \)[/tex], the constant [tex]\( c \)[/tex] shifts the function vertically.
As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( +\infty \)[/tex]), the term [tex]\( e^{-x} \)[/tex] approaches zero because the exponent [tex]\( -x \)[/tex] becomes a very large negative number, making [tex]\( e^{-x} \)[/tex] close to zero. Therefore, the function [tex]\( f(x) \)[/tex] approaches [tex]\( c \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( +\infty \)[/tex].
In our case, since [tex]\( c = 9 \)[/tex], the horizontal asymptote of the function [tex]\( f(x) = e^{-x} + 9 \)[/tex] is:
[tex]\[
y = 9
\][/tex]
2. Determine the Range:
To determine the range of [tex]\( f(x) = e^{-x} + 9 \)[/tex], examine the behavior of [tex]\( f(x) \)[/tex].
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( -\infty \)[/tex]), the term [tex]\( e^{-x} \)[/tex] approaches positive infinity, making the function [tex]\( f(x) = e^{-x} + 9 \)[/tex] approach positive infinity.
- As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( +\infty \)[/tex]), the term [tex]\( e^{-x} \)[/tex] approaches zero, and thus [tex]\( f(x) = e^{-x} + 9 \)[/tex] approaches 9.
Therefore, for any value of [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] is always greater than 9. More formally, [tex]\( f(x) \)[/tex] will take values greater than 9 but never less than 9 or exactly equal to 9 in real terms.
Hence, the range of [tex]\( f(x) = e^{-x} + 9 \)[/tex] is:
[tex]\[
(9, \infty)
\][/tex]
In conclusion:
- The asymptote is:
[tex]\[
y = 9
\][/tex]
- The range is:
[tex]\[
(9, \infty)
\][/tex]
