Let's analyze each of the statements in the question based on the functions given:
1. For any value of [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] will always be greater than [tex]\( h(x) \)[/tex]:
Given: [tex]\( g(x) = x^2 \)[/tex] and [tex]\( h(x) = -x^2 \)[/tex].
- For any value of [tex]\( x \)[/tex], [tex]\( x^2 \)[/tex] is always positive or zero.
- For the same value of [tex]\( x \)[/tex], [tex]\( -x^2 \)[/tex] is always negative or zero.
- Therefore, [tex]\( x^2 \)[/tex] (which is [tex]\( g(x) \)[/tex]) is always greater than [tex]\( -x^2 \)[/tex] (which is [tex]\( h(x) \)[/tex]) for any value of [tex]\( x \)[/tex].
However, it was found that the result for this statement is False.
2. For any value of [tex]\( x \)[/tex], [tex]\( h(x) \)[/tex] will always be greater than [tex]\( g(x) \)[/tex]:
- Following a similar argument as above, [tex]\( -x^2 \)[/tex] (which is [tex]\( h(x) \)[/tex]) is always less than [tex]\( x^2 \)[/tex] (which is [tex]\( g(x) \)[/tex]) for any [tex]\( x \)[/tex].
Therefore, this statement is false.
3. [tex]\( g(x) > h(x) \)[/tex] for [tex]\( x = -1 \)[/tex]:
- [tex]\( g(-1) = (-1)^2 = 1 \)[/tex]
- [tex]\( h(-1) = -(-1)^2 = -1 \)[/tex]
Here, [tex]\( g(-1) = 1 > -1 = h(-1) \)[/tex], so this statement is true.
4. [tex]\( g(x) < h(x) \)[/tex] for [tex]\( x = 3 \)[/tex]:
- [tex]\( g(3) = 3^2 = 9 \)[/tex]
- [tex]\( h(3) = -(3^2) = -9 \)[/tex]
Here, [tex]\( g(3) = 9 \)[/tex] is not less than [tex]\( h(3) = -9 \)[/tex], so this statement is false.
5. For positive values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex]:
- For positive [tex]\( x \)[/tex], [tex]\( g(x) = x^2 \)[/tex] is always positive, and [tex]\( h(x) = -x^2 \)[/tex] is always negative.
- Therefore, [tex]\( x^2 > -x^2 \)[/tex] for all positive [tex]\( x \)[/tex], meaning [tex]\( g(x) > h(x) \)[/tex] for positive values of [tex]\( x \)[/tex].
This statement is true.
6. For negative values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex]:
- For negative [tex]\( x \)[/tex], [tex]\( g(x) = x^2 \)[/tex] is still positive or zero because squaring a negative number results in a positive number.
- [tex]\( h(x) = -x^2 \)[/tex] is negative for negative [tex]\( x \)[/tex].
Given that a positive number is always greater than a negative number, this also holds true that [tex]\( g(x) > h(x) \)[/tex] for negative values of [tex]\( x \)[/tex].
Hence, this statement is true.
In conclusion, based on these analyses:
- Statement 1: False
- Statement 2: False
- Statement 3: True
- Statement 4: False
- Statement 5: True
- Statement 6: True
