Answer :
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.
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.
