To determine the horizontal asymptote for the function [tex]\( f(x) = e^x + 7 \)[/tex]:
1. The exponential function [tex]\( e^x \)[/tex] approaches zero as [tex]\( x \)[/tex] approaches negative infinity.
2. Therefore, the function [tex]\( f(x) = e^x + 7 \)[/tex] will approach [tex]\( 7 \)[/tex] as [tex]\( x \)[/tex] approaches negative infinity.
3. Hence, the horizontal asymptote of the function is given by the equation [tex]\( y = 7 \)[/tex].
Next, let's determine the range of the function:
1. The function [tex]\( e^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex] and its values range from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex].
2. By adding 7 to [tex]\( e^x \)[/tex], the smallest value [tex]\( f(x) \)[/tex] can take is [tex]\( 0 + 7 = 7 \)[/tex] and it can increase without bound as [tex]\( x \)[/tex] increases.
3. Hence, the range of the function [tex]\( f(x) = e^x + 7 \)[/tex] is from [tex]\( 7 \)[/tex] (exclusive) to [tex]\( \infty \)[/tex] (infinity).
In interval notation, the range is [tex]\( (7, \infty) \)[/tex].
To summarize:
- The asymptote of the function [tex]\( f(x) = e^x + 7 \)[/tex] is [tex]\( y = 7 \)[/tex].
- The range of the function is [tex]\( (7, \infty) \)[/tex].
Thus, the asymptote is [tex]\( y = 7 \)[/tex], and the range is [tex]\( (7, \infty) \)[/tex].
