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A function is translated from [tex]$f(x)=2^x$[/tex] to [tex]$g(x)=2^{x-3}$[/tex]. What is the effect on [tex][tex]$f(x)$[/tex][/tex]?

A. [tex]$f(x)$[/tex] moves 3 units upward.
B. [tex]$f(x)$[/tex] moves 3 units to the left.
C. [tex][tex]$f(x)$[/tex][/tex] moves 3 units to the right.
D. [tex]$f(x)$[/tex] moves 3 units downward.

Sagot :

GinnyAnswer
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.
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