The problem asks about the function $f(x) = 3^{x-1} + 5$ and how its graph compares to the graph of $g(x) = 3^x$.

Analysis:

1. Function $f(x) = 3^{x-1} + 5$:

   • The function $f(x)$ is a transformation of the basic exponential function $g(x) = 3^x$.
   • The expression $x-1$ inside the exponent represents a horizontal shift. Since $x-1$ is the same as $x - 1$, it indicates a shift to the right by 1 unit.
   • The +5 outside the exponential term indicates a vertical shift of 5 units upwards.

2. Function $g(x) = 3^x$:

   • This is the standard exponential function with a base of 3.
   • The graph of this function passes through the point $(0, 1)$ and increases rapidly as $x$ increases.

Comparison:

• The graph of $f(x)$ will look very similar to the graph of $g(x)$, but:
  1. It will be shifted 1 unit to the right (because of $x-1$).
  2. It will be shifted 5 units upward (due to the +5).

Conclusion:

The graph of $f(x) = 3^{x-1} + 5$ is a transformation of $g(x) = 3^x$, shifted 1 unit to the right and 5 units upward.

Would you like more details about the transformations or have any questions about how this applies to other functions?

Here are some related questions for further exploration:

1. How would the graph change if the shift was to the left instead of to the right?
2. How would the graph behave if the vertical shift was down instead of up?
3. What happens to the graph if the exponent is increased by a larger constant, such as $3^{x-2} + 5$?
4. How would the graph change if we used a base other than 3, like $2^x$?
5. What effect does changing the vertical shift have on the asymptote of the function?

Tip: When analyzing shifts in functions, remember that horizontal shifts involve changes inside the function, while vertical shifts affect the function outside the exponent.