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Sagot :
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]
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]
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