To understand the difference between the graphs of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], let us first look at their definitions more closely:
1. The function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[
f(x) = \frac{2}{3}x^3 + 1
\][/tex]
2. The function [tex]\( g(x) \)[/tex] is defined as:
[tex]\[
g(x) = \frac{2}{3}(-x)^3 + 1
\][/tex]
Next, we simplify the expression for [tex]\( g(x) \)[/tex]:
- Note that [tex]\((-x)^3 = -x^3\)[/tex]. Therefore, we can rewrite [tex]\( g(x) \)[/tex] as:
[tex]\[
g(x) = \frac{2}{3}(-x)^3 + 1 = \frac{2}{3}(-x^3) + 1 = -\frac{2}{3}x^3 + 1
\][/tex]
The transformation from [tex]\( f(x) = \frac{2}{3}x^3 + 1 \)[/tex] to [tex]\( g(x) = -\frac{2}{3}x^3 + 1 \)[/tex] involves negating the [tex]\( x \)[/tex]-term inside the cubic function. This operation is indicative of a reflection over the [tex]\( y \)[/tex]-axis.
To confirm this, consider what reflecting a point [tex]\( (x, y) \)[/tex] over the [tex]\( y \)[/tex]-axis means:
- The point [tex]\( (x, y) \)[/tex] becomes [tex]\( (-x, y) \)[/tex].
In the context of functions, replacing [tex]\( x \)[/tex] with [tex]\(-x\)[/tex] in [tex]\( f(x) \)[/tex] yields the function [tex]\( g(x) \)[/tex], demonstrating that [tex]\( g(x) \)[/tex] is indeed [tex]\( f(x) \)[/tex] reflected over the [tex]\( y \)[/tex]-axis.
Therefore, the correct description of the difference between the graphs of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is:
A. [tex]\( g(x) \)[/tex] is a reflection of [tex]\( f(x) \)[/tex] over the [tex]\( y \)[/tex]-axis.
