To determine the relationship between [tex]\( F(-2) \)[/tex] and [tex]\( F(2) \)[/tex] given that [tex]\( F \)[/tex] is a decreasing function, we should recall the definition and properties of decreasing functions.
1. Definition of a Decreasing Function: A function [tex]\( F \)[/tex] is said to be decreasing if, for any two real numbers [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] such that [tex]\( x_1 < x_2 \)[/tex], it holds that [tex]\( F(x_1) \geq F(x_2) \)[/tex]. If a function is strictly decreasing, then [tex]\( F(x_1) > F(x_2) \)[/tex] whenever [tex]\( x_1 < x_2 \)[/tex].
2. Applying the Definition: We are given that [tex]\( F \)[/tex] is a decreasing function. Let's analyze the specific values of [tex]\( x = -2 \)[/tex] and [tex]\( x = 2 \)[/tex].
3. Comparison of [tex]\( -2 \)[/tex] and [tex]\( 2 \)[/tex]: Notice that:
[tex]\[
-2 < 2
\][/tex]
This relationship between [tex]\(-2\)[/tex] and [tex]\(2\)[/tex] fits into the definition of a decreasing function because [tex]\(-2\)[/tex] is less than [tex]\(2\)[/tex].
4. Conclusion about [tex]\( F(-2) \)[/tex] and [tex]\( F(2) \)[/tex]:
Based on the property of a decreasing function, if [tex]\( x_1 < x_2 \)[/tex], then [tex]\( F(x_1) \geq F(x_2) \)[/tex]. Substituting [tex]\( x_1 = -2 \)[/tex] and [tex]\( x_2 = 2 \)[/tex], we get:
[tex]\[
F(-2) \geq F(2)
\][/tex]
Since [tex]\( F \)[/tex] is strictly decreasing in the context mentioned in this problem, we actually have:
[tex]\[
F(-2) > F(2)
\][/tex]
Thus, considering the property of the decreasing function [tex]\( F \)[/tex] and the comparison between [tex]\(-2\)[/tex] and [tex]\(2\)[/tex], we conclude:
[tex]\[
\boxed{F(-2) > F(2)}
\][/tex]
