Let's analyze each statement one by one for the functions [tex]\( g(x) = x^2 \)[/tex] and [tex]\( h(x) = -x^2 \)[/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]
- [tex]\( g(x) = x^2 \)[/tex] will be greater than [tex]\( h(x) = -x^2 \)[/tex] for any value of [tex]\( x \)[/tex].
- Result: False
2. For any value of [tex]\( x \)[/tex], [tex]\( h(x) \)[/tex] will always be greater than [tex]\( g(x) \)[/tex]:
- [tex]\( g(x) = x^2 \)[/tex]
- [tex]\( h(x) = -x^2 \)[/tex]
- [tex]\( -x^2 \)[/tex] (which is [tex]\( h(x) \)[/tex]) will never be greater than [tex]\( x^2 \)[/tex] (which is [tex]\( g(x) \)[/tex]) for any value of [tex]\( x \)[/tex].
- Result: 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]
- [tex]\( 1 > -1 \)[/tex]
- Result: 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]
- [tex]\( 9 < -9 \)[/tex] is not true
- Result: False
5. For positive values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex]:
- For [tex]\( x > 0 \)[/tex]:
- [tex]\( g(x) = x^2 \)[/tex]
- [tex]\( h(x) = -x^2 \)[/tex]
- [tex]\( x^2 \)[/tex] will always be greater than [tex]\(-x^2 \)[/tex] for any positive [tex]\( x \)[/tex].
- Result: True
6. For negative values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex]:
- For [tex]\( x < 0 \)[/tex]:
- [tex]\( g(x) = x^2 \)[/tex]
- [tex]\( h(x) = -x^2 \)[/tex]
- [tex]\( x^2 \)[/tex] will always be greater than [tex]\(-x^2 \)[/tex] for any negative [tex]\( x \)[/tex].
- Result: True
Based on the analysis, these are the results:
- For any value of [tex]\( x, g(x) \)[/tex] will always be greater than [tex]\( h(x) \)[/tex]: False
- For any value of [tex]\( x, h(x) \)[/tex] will always be greater than [tex]\( g(x) \)[/tex]: False
- [tex]\( g(x) > h(x) \)[/tex] for [tex]\( x = -1 \)[/tex]: True
- [tex]\( g(x) < h(x) \)[/tex] for [tex]\( x = 3 \)[/tex]: False
- For positive values of [tex]\( x, g(x) > h(x) \)[/tex]: True
- For negative values of [tex]\( x, g(x) > h(x) \)[/tex]: True
