Given the functions [tex]\( g(x) = x^2 \)[/tex] and [tex]\( h(x) = -x^2 \)[/tex], let's analyze the truthfulness of each statement step by step:
1. For any value of [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] will always be greater than [tex]\( h(x) \)[/tex]:
- [tex]\( g(x) = x^2 \)[/tex]
- [tex]\( h(x) = -x^2 \)[/tex]
If we compare [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex], we get:
[tex]\[
x^2 > -x^2
\][/tex]
This translates to:
[tex]\[
2x^2 > 0 \quad \text{which is always true for all \( x \)}.
\][/tex]
Therefore, this statement is true.
2. For any value of [tex]\( x \)[/tex], [tex]\( h(x) \)[/tex] will always be greater than [tex]\( g(x) \)[/tex]:
Using the same functions:
[tex]\[
-x^2 > x^2 \quad \text{which is never true for any real \( x \)}
\][/tex]
This statement is false.
3. [tex]\( g(x) > h(x) \)[/tex] for [tex]\( x = -1 \)[/tex]:
Evaluating the functions at [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = (-1)^2 = 1
\][/tex]
[tex]\[
h(-1) = -(-1)^2 = -1
\][/tex]
Clearly,
[tex]\[
1 > -1
\][/tex]
Thus, this statement is true.
4. [tex]\( g(x) < h(x) \)[/tex] for [tex]\( x = 3 \)[/tex]:
Evaluating the functions at [tex]\( x = 3 \)[/tex]:
[tex]\[
g(3) = 3^2 = 9
\][/tex]
[tex]\[
h(3) = -(3)^2 = -9
\][/tex]
Clearly,
[tex]\[
9 > -9
\][/tex]
This statement is false because [tex]\( g(x) \)[/tex] is not less than [tex]\( h(x) \)[/tex]; it is greater.
5. For positive values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex]:
For [tex]\( x > 0 \)[/tex]:
[tex]\[
g(x) = x^2 \quad \text{and} \quad h(x) = -x^2
\][/tex]
It is clear that:
[tex]\[
x^2 > -x^2
\][/tex]
Hence for all positive [tex]\( x \)[/tex]:
[tex]\[
g(x) > h(x)
\][/tex]
Therefore, this statement is true.
6. For negative values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex]:
For [tex]\( x < 0 \)[/tex]:
[tex]\[
g(x) = x^2 \quad \text{and} \quad h(x) = -x^2
\][/tex]
Even for negative [tex]\( x \)[/tex]:
[tex]\[
x^2 > -x^2
\][/tex]
Hence for all negative [tex]\( x \)[/tex]:
[tex]\[
g(x) > h(x)
\][/tex]
Therefore, this statement is true.
Summarizing the true statements:
- [tex]\( g(x) > h(x) \)[/tex] for [tex]\( x = -1 \)[/tex] (Statement 3)
- For positive values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex] (Statement 5)
- For negative values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex] (Statement 6)
Thus, the true statements are:
[tex]\( \boxed{3, 5, 6} \)[/tex]
