the graph shows that ( f(x) = 3^{x} ) is translated horizontally and vertically to create the function (…

Explanation:

Step1: Recall the horizontal translation rule

For the function (y = a^{x - h}+k), if (h>0), the graph of (y = a^{x}) is translated (h) units to the right; if (h < 0), the graph is translated (|h|) units to the left. The original function (f(x)=3^{x}) passes through the point ((0,1)) (since (f(0)=3^{0}=1)).

Step2: Identify a key - point on (g(x))

Let's find a key - point on (g(x)). We know that (g(x)=3^{x - h}+k). Suppose we use the horizontal translation. If we consider the point ((2,3)) on (g(x)) (by looking at the graph). Substitute (x = 2) into (g(x)): (g(2)=3^{2 - h}+k). For (f(x)=3^{x}), when (x = 0), (y = 1). The horizontal translation formula: if the point ((x_0,y_0)) on (y = f(x)) is translated to ((x_1,y_1)) on (y = f(x - h)+k), then (x_1=x_0 + h). We know that the horizontal shift: if we consider the "basic" point of (y = 3^{x}) (where (y = 1) when (x = 0)) and the corresponding point on (y=3^{x - h}+k) (let's assume (k = 1) for the vertical part, but focusing on the horizontal). The graph of (y = 3^{x}) is shifted to the right. If we use the formula (x) (for (g(x))) (=x) (for (f(x))) (+h). Take a point: for (f(x)=3^{x}), when (y = 2), (x=\log_3{2}\approx0.63) (not the best approach). Another way: The general form of horizontal translation. The function (y = 3^{x}) is transformed to (y = 3^{x - h}). The (y) - intercept of (y = 3^{x}) is at (x = 0), and for (y=3^{x - h}), when (y = 2) (a value on the (y) - axis for (g(x)) near the (y) - intercept - like behavior, assume (k = 1) is a vertical shift. If we consider the fact that the graph of (y = 3^{x}) is shifted (2) units to the right. Using the formula (y = f(x - h)) (where (f(x)=3^{x})), when (h = 2), (y = 3^{x - 2}).

Answer:

(2)