Answer:
The required function is
.
Step-by-step explanation:
The given parent function is
![f(x)=5^x](https://tex.z-dn.net/?f=f%28x%29%3D5%5Ex)
The transformation of a parent function is defined as
![g(x)=kf(x+a)+b](https://tex.z-dn.net/?f=g%28x%29%3Dkf%28x%2Ba%29%2Bb)
Where, k represents the vertical stretch or compression, a represents the horizontal shift and b represent the vertical shift.
→ If |k|>1, then it represents vertically stretch and If |k|<1, then it represent vertically compression.
→ If a>0, then f(x) shifts left by a units and If a<0, then f(x) shifts right by a units.
→If b>0, then f(x) shifts upward by b units and If b<0, then f(x) shifts downward by b units.
Since the graph of f(x) has been vertically compressed by a factor of one-half shifted to the right three units and down two units.
![k=\frac{1}{2}](https://tex.z-dn.net/?f=k%3D%5Cfrac%7B1%7D%7B2%7D)
![a=-3](https://tex.z-dn.net/?f=a%3D-3)
![b=-2](https://tex.z-dn.net/?f=b%3D-2)
The graph of required function is
![g(x)=\frac{1}{2}f(x-3)-2](https://tex.z-dn.net/?f=g%28x%29%3D%5Cfrac%7B1%7D%7B2%7Df%28x-3%29-2)
![g(x)=\frac{1}{2}(5^{x-3})-2](https://tex.z-dn.net/?f=g%28x%29%3D%5Cfrac%7B1%7D%7B2%7D%285%5E%7Bx-3%7D%29-2)
Therefore the required function is
.