Certainly! Let’s break down how the graph of the function [tex]\( g(x) \)[/tex] will differ from the graph of the function [tex]\( f(x) \)[/tex].
First, let's recognize the given functions:
[tex]\[ f(x) = 2^x \][/tex]
[tex]\[ g(x) = f(x) + 6 \][/tex]
By substituting the expression of [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 2^x + 6 \][/tex]
To understand how the graph of [tex]\( g(x) \)[/tex] differs from the graph of [tex]\( f(x) \)[/tex], we need to observe the transformation applied through the term [tex]\( + 6 \)[/tex].
The function [tex]\( g(x) \)[/tex] does precisely this: it takes each value of [tex]\( f(x) \)[/tex] and adds 6 to it. Graphically, adding a constant to a function [tex]\( f(x) \)[/tex] will shift the graph of [tex]\( f(x) \)[/tex] vertically.
Here’s the step-by-step reasoning:
1. Consider a point [tex]\( (a, f(a)) \)[/tex] on the graph of [tex]\( f(x) \)[/tex]. This point corresponds to [tex]\( (a, 2^a) \)[/tex].
2. For the same value [tex]\( a \)[/tex], the corresponding point on the graph of [tex]\( g(x) \)[/tex] will be [tex]\( (a, g(a)) \)[/tex] which is [tex]\( (a, 2^a + 6) \)[/tex].
Notice that for every point [tex]\( a \)[/tex], [tex]\( g(a) = f(a) + 6 \)[/tex]. This means each point on the graph of [tex]\( f(x) \)[/tex] is shifted up by 6 units to get the graph of [tex]\( g(x) \)[/tex].
Thus, the correct description is:
- The graph of the function [tex]\( g(x) \)[/tex] is the graph of the function [tex]\( f(x) \)[/tex] shifted 6 units up.
So, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
This means the solution is option [tex]\( C \)[/tex]: The graph of function [tex]\( g \)[/tex] is the graph of function [tex]\( f \)[/tex] shifted 6 units up.
