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

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

Sagot :

GinnyAnswer
Let's analyze each statement 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].
- To determine if [tex]\(g(x) > h(x)\)[/tex] for any value of [tex]\(x\)[/tex], we need to see if [tex]\(x^2 > -x^2\)[/tex]. This is equivalent to [tex]\(2x^2 > 0\)[/tex], which is true for any non-zero [tex]\(x\)[/tex]. Therefore, this statement is true.
- However, since we only need [tex]\(x \neq 0\)[/tex] for the inequality to be true, [tex]\(g(0)= h(0)=0\)[/tex], the inequality is not strict for [tex]\(x=0\)[/tex], thus generally considered true.

2. For any value of [tex]\(x\)[/tex], [tex]\(h(x)\)[/tex] will always be greater than [tex]\(g(x)\)[/tex].
- Similarly, we need to check if [tex]\(-x^2 > x^2\)[/tex], which would simplify to [tex]\(0 > 2x^2\)[/tex]. This is false for any real [tex]\(x\)[/tex] except at [tex]\(x = 0\)[/tex], but for any non-zero [tex]\(x\)[/tex], this inequality will not hold and thus this statement is false.

3. [tex]\(g(x) > h(x)\)[/tex] for [tex]\(x = -1\)[/tex].
- First, compute [tex]\(g(-1) = (-1)^2 = 1\)[/tex] and [tex]\(h(-1) = -(-1)^2 = -1\)[/tex].
- Therefore, [tex]\(g(-1) > h(-1)\)[/tex] is [tex]\(1 > -1\)[/tex], which is true.

4. [tex]\(g(x) < h(x)\)[/tex] for [tex]\(x = 3\)[/tex].
- Compute [tex]\(g(3) = 3^2 = 9\)[/tex] and [tex]\(h(3) = -3^2 = -9\)[/tex].
- So, [tex]\(g(3) < h(3)\)[/tex] is [tex]\(9 < -9\)[/tex], which is clearly 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] and [tex]\(h(x) = -x^2\)[/tex].
- Here, [tex]\(x^2 > -x^2\)[/tex], which holds for any positive [tex]\(x\)[/tex], thus this statement is true.

6. For negative values of [tex]\(x\)[/tex], [tex]\(g(x) > h(x)\)[/tex].
- For [tex]\(x < 0\)[/tex], [tex]\(g(x) = x^2\)[/tex] and [tex]\(h(x) = -x^2\)[/tex].
- Since [tex]\(x^2 > -x^2\)[/tex], the same logic applies, making this statement true for any negative [tex]\(x\)[/tex].

Given these considerations:
- Statements 1, 3, 5, and 6 are true.
- Statements 2 and 4 are false.
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