To solve this problem, we need to understand how transformations affect the graph of a function. We're given the function [tex]\( f(x) = 10^x \)[/tex] and the function [tex]\( g(z) \)[/tex] expressed as [tex]\( g(z) = f(z-6) = 10^{(x-6)} \)[/tex].
When we have a function [tex]\( f(x) \)[/tex] and we replace [tex]\( x \)[/tex] with [tex]\( (x - c) \)[/tex] to get a new function [tex]\( f(x - c) \)[/tex], the graph of the new function is shifted horizontally. Specifically:
- If [tex]\( c \)[/tex] is positive, the graph is shifted to the right by [tex]\( c \)[/tex] units.
- If [tex]\( c \)[/tex] is negative, the graph is shifted to the left by [tex]\( |c| \)[/tex] units.
In [tex]\( g(x) = 10^{(x-6)} \)[/tex], the expression [tex]\( (x - 6) \)[/tex] indicates a horizontal transformation where [tex]\( c = 6 \)[/tex]. Therefore:
- The function [tex]\( g(x) \)[/tex] is the function [tex]\( f(x) \)[/tex] shifted to the right by 6 units.
So, the correct interpretation is that the graph of [tex]\( g(x) = 10^{(x-6)} \)[/tex] is the graph of [tex]\( f(x) = 10^x \)[/tex] shifted to the right by 6 units.
Hence, the correct answer is:
C. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] shifted to the right 6 units.
