Answer:
There is no exponential function passing through (1,1) and (5,5).
Step-by-step explanation:
We have the following exponential function
![y = Ce^{kt}](https://tex.z-dn.net/?f=y%20%3D%20Ce%5E%7Bkt%7D)
The function passes through these two points:
(1,1): This means that when t = 1, y = 1
(5,5): This means that when t = 5, y = 5.
So
(1,1)
![y = Ce^{kt}](https://tex.z-dn.net/?f=y%20%3D%20Ce%5E%7Bkt%7D)
![Ce^{k} = 1](https://tex.z-dn.net/?f=Ce%5E%7Bk%7D%20%3D%201)
![e^{k} = \frac{1}{C}](https://tex.z-dn.net/?f=e%5E%7Bk%7D%20%3D%20%5Cfrac%7B1%7D%7BC%7D)
(5,5)
![y = Ce^{kt}](https://tex.z-dn.net/?f=y%20%3D%20Ce%5E%7Bkt%7D)
![5Ce^{k} = 1](https://tex.z-dn.net/?f=5Ce%5E%7Bk%7D%20%3D%201)
![Ce^{k} = \frac{1}{5}](https://tex.z-dn.net/?f=Ce%5E%7Bk%7D%20%3D%20%5Cfrac%7B1%7D%7B5%7D)
From above, we have that:
![e^{k} = \frac{1}{C}](https://tex.z-dn.net/?f=e%5E%7Bk%7D%20%3D%20%5Cfrac%7B1%7D%7BC%7D)
![C\frac{1}{C} = \frac{1}{5}](https://tex.z-dn.net/?f=C%5Cfrac%7B1%7D%7BC%7D%20%3D%20%5Cfrac%7B1%7D%7B5%7D)
![1 = \frac{1}{5}](https://tex.z-dn.net/?f=1%20%3D%20%5Cfrac%7B1%7D%7B5%7D)
1 cannot be equal to 1/5, so this is wrong.
This means that there is no exponential function passing through (1,1) and (5,5).