To determine how the graph of [tex]\( g(x) \)[/tex] differs from the graph of [tex]\( f(x) \)[/tex], let's examine the given functions:
[tex]\[ f(x) = 2^x \][/tex]
[tex]\[ g(x) = f(x) + 4 = 2^x + 4 \][/tex]
The function [tex]\( g(x) \)[/tex] takes the original function [tex]\( f(x) \)[/tex] and adds 4 to it. This addition is applied to the output of [tex]\( f(x) \)[/tex], meaning that for any input [tex]\( x \)[/tex], the output of [tex]\( g(x) \)[/tex] is 4 units greater than the output of [tex]\( f(x) \)[/tex].
This can be visualized as a vertical shift:
- Adding a constant to a function results in a vertical shift of the graph.
- In this case, [tex]\( f(x) \)[/tex] is shifted vertically by [tex]\( +4 \)[/tex].
Therefore, the graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] shifted 4 units upward.
The correct answer is:
D. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] shifted 4 units upward.
