Answer:
The exponential decay function is
⇒ 3rd answer
Step-by-step explanation:
* Lets explain the exponential decay function
- The general form of an exponential Function is
,
where a is the initial value and b is growth factor
- If b > 1 , then the function is exponential growth function
- If 0 < b < 1 , then the function is exponential decay function
- The function
can be written as
![f(x)=a(\frac{1}{b})^{x}](https://tex.z-dn.net/?f=f%28x%29%3Da%28%5Cfrac%7B1%7D%7Bb%7D%29%5E%7Bx%7D)
# <em>Remember</em>: if 0 < b < 1 , then 1/b > 1 , then change the negative
sign of the power by reciprocal the growth factor to decide the
function is growth or decay
* Lets solve the problem
1. ![f(x)=\frac{3}{4}(\frac{7}{4})^{x}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B3%7D%7B4%7D%28%5Cfrac%7B7%7D%7B4%7D%29%5E%7Bx%7D)
∵ b = 7/4
∵ 7/4 is greater than 1
∴ f(x) is an exponential growth function
2. ![f(x)=\frac{2}{3}(\frac{4}{5})^{-x}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B2%7D%7B3%7D%28%5Cfrac%7B4%7D%7B5%7D%29%5E%7B-x%7D)
- Change the (-x) to x by reciprocal (4/5) to (5/4)
∴ ![f(x)=\frac{2}{3}(\frac{5}{4})^{x}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B2%7D%7B3%7D%28%5Cfrac%7B5%7D%7B4%7D%29%5E%7Bx%7D)
∵ b = 5/4
∵ 5/4 is greater than 1
∴ f(x) is an exponential growth function
3. ![f(x)=\frac{3}{2}(\frac{8}{7})^{-x}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B3%7D%7B2%7D%28%5Cfrac%7B8%7D%7B7%7D%29%5E%7B-x%7D)
- Change the (-x) to x by reciprocal (8/7) to (7/8)
∴ ![f(x)=\frac{3}{2}(\frac{7}{8})^{x}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B3%7D%7B2%7D%28%5Cfrac%7B7%7D%7B8%7D%29%5E%7Bx%7D)
∵ b = 7/8
∵ 7/8 is greater than 0 and smaller than 1 ⇒ 0 < 7/8 < 1
∴ f(x) is an exponential decay function
* The exponential decay function is ![f(x)=\frac{3}{2}(\frac{8}{7})^{-x}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B3%7D%7B2%7D%28%5Cfrac%7B8%7D%7B7%7D%29%5E%7B-x%7D)