To reflect the graph of the function $y = 3^x + 2$ with respect to the line $y = x$, we need to find the inverse of the function $y = 3^x + 2$, as reflecting over the line $y = x$ swaps the roles of $x$ and $y$.

Steps:

1. Start with the original equation:
   $y = 3^x + 2$

2. Swap $x$ and $y$ to reflect over the line $y = x$:
   $x = 3^y + 2$

3. Solve for $y$ (which will now be the equation of the reflected function): $x - 2 = 3^y$ Take the natural logarithm of both sides: $\ln(x - 2) = \ln(3^y)$ Using logarithmic properties: $\ln(x - 2) = y \ln(3)$ Solve for $y$: $y = \frac{\ln(x - 2)}{\ln(3)}$

Thus, the equation of the reflected graph is: $y = \frac{\ln(x - 2)}{\ln(3)}$

Final Answer:

The reflection of $y = 3^x + 2$ with respect to the line $y = x$ is: $y = \frac{\ln(x - 2)}{\ln(3)}$

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Would you like a detailed explanation of any part of this process? Here are five related questions to expand your understanding:

1. How do we find the inverse of exponential functions in general?
2. What are the properties of logarithms that help solve such equations?
3. How does reflecting a function over the line $y = x$ relate to finding its inverse?
4. What does the graph of $y = 3^x + 2$ look like before and after reflection?
5. How do vertical and horizontal shifts affect reflections over the line $y = x$?

Tip: Reflecting over the line $y = x$ is the same as finding the inverse of a function. Always remember to swap $x$ and $y$ and then solve for $y$.