Let's carefully analyze the given functions:
1. [tex]\( f(x) = x^3 \)[/tex]
2. [tex]\( g(x) = (x + 1)^3 \)[/tex]
We are given that [tex]\( g(x) \)[/tex] is a transformation of [tex]\( f(x) \)[/tex]. Specifically, we need to determine what kind of transformation it is.
Let's rewrite [tex]\( g(x) \)[/tex] to better compare it to [tex]\( f(x) \)[/tex]:
- For [tex]\( g(x) = (x + 1)^3 \)[/tex], notice how the expression inside the parenthesis, [tex]\(x + 1\)[/tex], shifts the input of the function [tex]\(x\)[/tex] horizontally.
To understand what kind of shift this is, remember the general form of transformations:
- A horizontal shift occurs when we replace [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex] with [tex]\( x + c \)[/tex]. If [tex]\( c \)[/tex] is positive, the graph shifts to the left by [tex]\( c \)[/tex] units. If [tex]\( c \)[/tex] is negative, the graph shifts to the right by [tex]\( |c| \)[/tex] units.
In this case:
- [tex]\( g(x) = f(x + 1) \)[/tex] implies [tex]\( c = 1 \)[/tex], therefore, the graph is shifted horizontally to the left by 1 unit.
Thus, the correct description of the graph of [tex]\( g(x) \)[/tex] is:
B. a horizontal transformation of [tex]\( f(x) \)[/tex] 1 unit to the left
This transformation translates the entire graph of the function [tex]\( f(x) \)[/tex] by moving every point 1 unit to the left on the x-axis.
