Answer:
A. They have the same end behavior as x approaches -infinity but different end behavior as x approaches infinity
Step-by-step explanation:
You want to compare the end behaviors of two decreasing exponential functions with different y-intercepts and different horizontal asymptotes.
Exponential function
An unmodified exponential function will tend to infinity as x approaches infinity. It is increasing everywhere. Such a function will approach a horizontal asymptote as x approaches negative infinity.
An exponential function can be made to be a decreasing function by either of ...
- reflection across the x-axis
- reflection across the y-axis.
End behavior
The function f(x) shown in the graph has been reflected across the y-axis and translated downward about 4 units. Hence it tends to infinity as x approaches negative infinity.
The function g(x) has a positive y-intercept and no x-intercept, hence it, too, must be reflected across the y-axis. Therefore, g(x), like f(x), will tend to infinity as x approaches negative infinity.
The fact that g(x) does not cross the x-axis means it has a different horizontal asymptote from f(x). The reflected functions will approach their respective (different) asymptotes as x approaches infinity. Hence, their right-end behavior is different.
In short, the two functions have the same end behavior as x approaches -infinity but different end behavior as x approaches infinity.
