To explore the key feature of the function [tex]\( g(x) = e^x - 7 \)[/tex], let's start by understanding the graph of the original function [tex]\( f(x) = e^x \)[/tex].
1. Function [tex]\( f(x) = e^x \)[/tex]:
- Horizontal Asymptote: The function [tex]\( f(x) = e^x \)[/tex] has a horizontal asymptote at [tex]\( y = 0 \)[/tex].
- Range: The range of [tex]\( f(x) = e^x \)[/tex] is [tex]\( (0, \infty) \)[/tex].
- Domain: The domain of [tex]\( f(x) = e^x \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
- y-intercept: The y-intercept of [tex]\( f(x) = e^x \)[/tex] occurs at the point (0, 1) because [tex]\( e^0 = 1 \)[/tex].
2. Transformation to [tex]\( g(x) = e^x - 7 \)[/tex]:
- The function [tex]\( g(x) = e^x - 7 \)[/tex] is obtained by shifting the graph of [tex]\( f(x) = e^x \)[/tex] downward by 7 units.
3. Key Features of [tex]\( g(x) = e^x - 7 \)[/tex]:
- Horizontal Asymptote: Since [tex]\( f(x) = e^x \)[/tex] has a horizontal asymptote at [tex]\( y = 0 \)[/tex], shifting the function downward by 7 units moves the horizontal asymptote from [tex]\( y = 0 \)[/tex] to [tex]\( y = -7 \)[/tex]. Therefore, the horizontal asymptote of [tex]\( g(x) \)[/tex] is [tex]\( y = -7 \)[/tex].
- Range: The range of [tex]\( g(x) = e^x - 7 \)[/tex] is [tex]\( (-7, \infty) \)[/tex] because for every value of [tex]\( e^x \)[/tex], subtracting 7 decreases its value by 7 units.
- Domain: The domain of [tex]\( g(x) = e^x - 7 \)[/tex] remains [tex]\( (-\infty, \infty) \)[/tex] because the function [tex]\( e^x \)[/tex] is defined for all real values of [tex]\( x \)[/tex], and shifting downward does not affect the domain.
- y-intercept: The y-intercept of [tex]\( g(x) = e^x - 7 \)[/tex] is found by evaluating [tex]\( g(0) \)[/tex]. Since [tex]\( g(0) = e^0 - 7 = 1 - 7 = -6 \)[/tex], the y-intercept is [tex]\((0, -6)\)[/tex].
Considering the above analysis, the key feature described by the correct statement is:
A. horizontal asymptote of [tex]\( y = -7 \)[/tex].
