Apologies for the confusion earlier. Let me re-analyze the functions and provide the correct solution.

Solution By Steps

Step 1: Analyze

f(x)=3\left(\frac{1}{2}\right)^{x}

This function represents a vertical stretch of the function

\left(\frac{1}{2}\right)^x by a factor of 3. It is a vertical stretch of the exponential function.

Step 2: Analyze

f(x)=\frac{1}{2}(3)^{x}

This function represents a vertical compression of the function

3^x by a factor of

\frac{1}{2}. It is not a vertical stretch.

Step 3: Analyze

f(x)=(3)^{2 x}

This function represents a horizontal stretch of the function

3^x by a factor of 2. It does not represent a vertical stretch.

Step 4: Analyze

f(x)=3^{\left(\frac{1}{2} x\right)}

This function represents a horizontal compression of the function

3^x by a factor of

\frac{1}{2}. It does not represent a vertical stretch.

Final Answer

The function

f(x)=3\left(\frac{1}{2}\right)^{x} represents a vertical stretch of an exponential function.