To find the equation of a function that has been shifted horizontally, we follow these steps:
1. Understand the Original Function: The original function given is [tex]\( f(x) = e^x \)[/tex].
2. Identify the Shift: The graph is shifted 6 units to the right.
3. Apply the Horizontal Shift: When a function [tex]\( f(x) \)[/tex] is shifted to the right by [tex]\( a \)[/tex] units, the new function [tex]\( g(x) \)[/tex] is given by [tex]\( g(x) = f(x - a) \)[/tex]. Here, "[tex]\( a \)[/tex]" is the number of units of the shift.
4. Substitute the Shift Value: In this case, [tex]\( a = 6 \)[/tex]. So, we need to shift [tex]\( f(x) = e^x \)[/tex] right by 6 units. Therefore, [tex]\( g(x) = f(x - 6) \)[/tex].
5. Formulate the New Function: Substitute [tex]\( x - 6 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
g(x) = e^{(x - 6)}
\][/tex]
Thus, the equation of the new graph is [tex]\( g(x) = e^{x - 6} \)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{g(x) = e^{x-6}} \][/tex]
