Find the domain and range for the following exponential function. $$ y=-e^{x-6}+5 $$ a) Record the horizontal asymptote below. Be sure to record you answer as $$ y= $$ \#. $$ y=5 $$ b) Record the domain below using interval notation. $$ (-\infty, \infty) $$ c) Record the range below using interval notation. $$ (-\infty, 5) $$

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- Horizontal asymptote: $$ y = 5 $$ - Domain: $$ (-\infty, \infty) $$ - Range: $$ (-\infty, 5) $$ Let's analyze the given exponential function $$ y = -e^{x-6} + 5 $$. ### a) Horizontal Asymptote The horizontal asymptote of an exponential function is determined by the constant term that the function approaches as $$ x $$ approaches infinity or negative infinity. In this case, as $$ x \to \infty $$, $$ e^{x-6} $$ approaches infinity, making $$ -e^{x-6} $$ approach negative infinity. Therefore, $$ y $$ approaches $$ 5 $$ from below. Thus, the horizontal asymptote is: $$ y = 5 $$ ### b) Domain The domain of an exponential function is all real numbers since you can input any real number into the function. Therefore, the domain in interval notation is: $$ (-\infty, \infty) $$ ### c) Range To find the range, we analyze the behavior of the function. The term $$ -e^{x-6} $$ will always be negative or zero, and as $$ x $$ increases, $$ -e^{x-6} $$ will decrease without bound. Thus, $$ y $$ will approach $$ 5 $$ but never actually reach it. Therefore, the range of the function is all values less than $$ 5 $$: $$ (-\infty, 5) $$ ### Summary - Horizontal asymptote: $$ y = 5 $$ - Domain: $$ (-\infty, \infty) $$ - Range: $$ (-\infty, 5) $$

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