Answer :
Let's carefully analyze each statement to determine whether it is true or false for the given 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] is always non-negative because squaring any real number results in a non-negative value.
- [tex]\( h(x) = -x^2 \)[/tex] is always non-positive because it is the negative of a non-negative value.
- Thus, for any real number [tex]\( x \)[/tex], [tex]\( x^2 \geq 0 \)[/tex] and [tex]\(-x^2 \leq 0 \)[/tex]. Therefore, [tex]\( x^2 > -x^2 \)[/tex] except when [tex]\( x = 0 \)[/tex] where [tex]\( g(x) = h(x) = 0 \)[/tex].
- This statement is True.
2. For any value of [tex]\( x \)[/tex], [tex]\( h(x) \)[/tex] will always be greater than [tex]\( g(x) \)[/tex]:
- As already established, [tex]\( h(x) = -x^2 \leq 0 \)[/tex] and [tex]\( g(x) = x^2 \geq 0 \)[/tex].
- Therefore, [tex]\( -x^2 \leq x^2 \)[/tex], and this will never be true for any non-zero [tex]\( x \)[/tex] since [tex]\( x^2 \geq 0 \)[/tex].
- This statement is False.
3. [tex]\( g(x) > h(x) \)[/tex] for [tex]\( x = -1 \)[/tex]:
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = (-1)^2 = 1 \][/tex]
[tex]\[ h(-1) = -(-1)^2 = -1 \][/tex]
- Clearly, [tex]\( g(-1) = 1 > -1 = h(-1) \)[/tex].
- This statement is True.
4. [tex]\( g(x) < h(x) \)[/tex] for [tex]\( x = 3 \)[/tex]:
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ g(3) = 3^2 = 9 \][/tex]
[tex]\[ h(3) = -(3^2) = -9 \][/tex]
- Clearly, [tex]\( g(3) = 9 \not< -9 = h(3) \)[/tex]; rather, [tex]\( 9 > -9 \)[/tex].
- This statement is False.
5. For positive values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex]:
- For any positive [tex]\( x \)[/tex]:
[tex]\[ g(x) = x^2 \geq 0 \][/tex]
[tex]\[ h(x) = -x^2 \leq 0 \][/tex]
- As already established, [tex]\( x^2 > -x^2 \)[/tex] except at [tex]\( x = 0 \)[/tex] which is not positive.
- This statement is True.
6. For negative values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex]:
- For any negative [tex]\( x \)[/tex]:
[tex]\[ g(x) = x^2 \geq 0 \][/tex]
[tex]\[ h(x) = -x^2 \leq 0 \][/tex]
- Similarly, [tex]\( x^2 > -x^2 \)[/tex] because squaring a negative number results in a positive value and its negative is a negative value.
- This statement is True.
So, the true statements are:
- For any value of [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] will always be greater than [tex]\( h(x) \)[/tex].
- [tex]\( g(x) > h(x) \)[/tex] for [tex]\( x = -1 \)[/tex].
- For positive values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex].
- For negative values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex].
Therefore, the final list of true statements is:
[tex]\[ \text{[True, False, True, False, True, True]} \][/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] is always non-negative because squaring any real number results in a non-negative value.
- [tex]\( h(x) = -x^2 \)[/tex] is always non-positive because it is the negative of a non-negative value.
- Thus, for any real number [tex]\( x \)[/tex], [tex]\( x^2 \geq 0 \)[/tex] and [tex]\(-x^2 \leq 0 \)[/tex]. Therefore, [tex]\( x^2 > -x^2 \)[/tex] except when [tex]\( x = 0 \)[/tex] where [tex]\( g(x) = h(x) = 0 \)[/tex].
- This statement is True.
2. For any value of [tex]\( x \)[/tex], [tex]\( h(x) \)[/tex] will always be greater than [tex]\( g(x) \)[/tex]:
- As already established, [tex]\( h(x) = -x^2 \leq 0 \)[/tex] and [tex]\( g(x) = x^2 \geq 0 \)[/tex].
- Therefore, [tex]\( -x^2 \leq x^2 \)[/tex], and this will never be true for any non-zero [tex]\( x \)[/tex] since [tex]\( x^2 \geq 0 \)[/tex].
- This statement is False.
3. [tex]\( g(x) > h(x) \)[/tex] for [tex]\( x = -1 \)[/tex]:
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = (-1)^2 = 1 \][/tex]
[tex]\[ h(-1) = -(-1)^2 = -1 \][/tex]
- Clearly, [tex]\( g(-1) = 1 > -1 = h(-1) \)[/tex].
- This statement is True.
4. [tex]\( g(x) < h(x) \)[/tex] for [tex]\( x = 3 \)[/tex]:
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ g(3) = 3^2 = 9 \][/tex]
[tex]\[ h(3) = -(3^2) = -9 \][/tex]
- Clearly, [tex]\( g(3) = 9 \not< -9 = h(3) \)[/tex]; rather, [tex]\( 9 > -9 \)[/tex].
- This statement is False.
5. For positive values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex]:
- For any positive [tex]\( x \)[/tex]:
[tex]\[ g(x) = x^2 \geq 0 \][/tex]
[tex]\[ h(x) = -x^2 \leq 0 \][/tex]
- As already established, [tex]\( x^2 > -x^2 \)[/tex] except at [tex]\( x = 0 \)[/tex] which is not positive.
- This statement is True.
6. For negative values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex]:
- For any negative [tex]\( x \)[/tex]:
[tex]\[ g(x) = x^2 \geq 0 \][/tex]
[tex]\[ h(x) = -x^2 \leq 0 \][/tex]
- Similarly, [tex]\( x^2 > -x^2 \)[/tex] because squaring a negative number results in a positive value and its negative is a negative value.
- This statement is True.
So, the true statements are:
- For any value of [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] will always be greater than [tex]\( h(x) \)[/tex].
- [tex]\( g(x) > h(x) \)[/tex] for [tex]\( x = -1 \)[/tex].
- For positive values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex].
- For negative values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex].
Therefore, the final list of true statements is:
[tex]\[ \text{[True, False, True, False, True, True]} \][/tex]
