Let's analyze the given functions to determine how the graph of [tex]\( g(x) \)[/tex] differs from the graph of [tex]\( f(x) \)[/tex].
1. We start with the function [tex]\( f(x) = 2^x \)[/tex].
2. The function [tex]\( g(x) \)[/tex] is defined as:
[tex]\[
g(x) = f(x + 4) = 2^{(x + 4)}
\][/tex]
3. When we compare [tex]\( g(x) \)[/tex] to [tex]\( f(x) \)[/tex]:
[tex]\[
g(x) = 2^{(x + 4)}
\][/tex]
This equation can be interpreted as the original function [tex]\( f(x) = 2^x \)[/tex] being modified by shifting the input [tex]\( x \)[/tex].
4. The new input for [tex]\( f(x) \)[/tex] is [tex]\( (x+4) \)[/tex]. This means that for every [tex]\( x \)[/tex] value, we now consider [tex]\( x + 4 \)[/tex] instead of just [tex]\( x \)[/tex].
5. Shifting the input [tex]\( x \)[/tex] by adding a constant results in a horizontal shift of the graph. Specifically, adding 4 to [tex]\( x \)[/tex] indicates a shift to the left by 4 units. In other words, each point on the graph of [tex]\( f(x) \)[/tex] moves 4 units to the left to match the graph of [tex]\( g(x) \)[/tex].
Therefore, the correct answer is:
D. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] shifted 4 units to the left.
