Answered

IDNLearn.com: Your trusted platform for finding precise and reliable answers. Our platform is designed to provide accurate and comprehensive answers to any questions you may have.

Evaluating More Than One Function

Which statements are true for the functions [tex][tex]$g(x)=x^2$[/tex][/tex] and [tex][tex]$h(x)=-x^2$[/tex][/tex]? Check all that apply.

A. For any value of [tex][tex]$x, g(x)$[/tex][/tex] will always be greater than [tex][tex]$h(x)$[/tex][/tex].
B. For any value of [tex][tex]$x, h(x)$[/tex][/tex] will always be greater than [tex][tex]$g(x)$[/tex][/tex].
C. [tex][tex]$g(x)\ \textgreater \ h(x)$[/tex][/tex] for [tex][tex]$x=-1$[/tex][/tex].
D. [tex][tex]$g(x)\ \textless \ h(x)$[/tex][/tex] for [tex][tex]$x=3$[/tex][/tex].
E. For positive values of [tex][tex]$x, g(x)\ \textgreater \ h(x)$[/tex][/tex].
F. For negative values of [tex][tex]$x, g(x)\ \textgreater \ h(x)$[/tex][/tex].

Sagot :

GinnyAnsweridn
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
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for choosing IDNLearn.com. We’re here to provide reliable answers, so please visit us again for more solutions.