To understand the effect on [tex]\( f(x) \)[/tex] when it is translated to form [tex]\( g(x) \)[/tex], let's analyze the transformation.
Given the functions:
[tex]\[ f(x) = 2^x \][/tex]
[tex]\[ g(x) = 2^{x-3} \][/tex]
The transformation involves changing the variable [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] to [tex]\( x-3 \)[/tex] in [tex]\( g(x) \)[/tex].
Horizontal shifts in the graphs of functions can be analyzed as follows:
- Replacing [tex]\( x \)[/tex] with [tex]\( x + c \)[/tex] shifts the graph [tex]\( c \)[/tex] units to the left.
- Replacing [tex]\( x \)[/tex] with [tex]\( x - c \)[/tex] shifts the graph [tex]\( c \)[/tex] units to the right.
Here, [tex]\( x \)[/tex] has been replaced with [tex]\( x - 3 \)[/tex], meaning the graph of [tex]\( f(x) \)[/tex] is shifted to the right by 3 units.
Therefore, the effect on [tex]\( f(x) \)[/tex] is:
[tex]\( f(x) \)[/tex] moves 3 units to the right.
