Get expert advice and insights on any topic with IDNLearn.com. Our experts are available to provide accurate, comprehensive answers to help you make informed decisions about any topic or issue you encounter.
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]
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.
