Let's analyze the statements given about 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 this, consider:
[tex]\[ g(x) = x^2 \][/tex]
[tex]\[ h(x) = -x^2 \][/tex]
For any [tex]\( x \in \mathbb{R} \)[/tex]:
[tex]\[ x^2 \geq 0 \][/tex]
[tex]\[ -x^2 \leq 0 \][/tex]
Since [tex]\( x^2 \)[/tex] is always non-negative and [tex]\(-x^2\)[/tex] is always non-positive, it is evident that [tex]\( x^2 \)[/tex] is always greater than [tex]\(-x^2\)[/tex] for any [tex]\( x \)[/tex].
Therefore, 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].
For any [tex]\( x \in \mathbb{R} \)[/tex]:
[tex]\[ g(x) = x^2 \][/tex]
[tex]\[ h(x) = -x^2 \][/tex]
As shown earlier:
[tex]\[ x^2 \geq 0 \][/tex]
[tex]\[ -x^2 \leq 0 \][/tex]
Hence, [tex]\( g(x) = x^2 \geq -x^2 = h(x) \)[/tex] and [tex]\( -x^2 \)[/tex] cannot be greater than [tex]\( x^2 \)[/tex] for any [tex]\( x \)[/tex].
Therefore, this statement is false.
3. [tex]\( g(x) > h(x) \)[/tex] for [tex]\( x = -1 \)[/tex].
Substitute [tex]\( x = -1 \)[/tex] into both functions:
[tex]\[ g(-1) = (-1)^2 = 1 \][/tex]
[tex]\[ h(-1) = -(-1)^2 = -1 \][/tex]
Clearly:
[tex]\[ g(-1) = 1 \][/tex]
[tex]\[ h(-1) = -1 \][/tex]
[tex]\[ 1 > -1 \][/tex]
Therefore, this statement is true.
4. [tex]\( g(x) < h(x) \)[/tex] for [tex]\( x = 3 \)[/tex].
Substitute [tex]\( x = 3 \)[/tex] into both functions:
[tex]\[ g(3) = 3^2 = 9 \][/tex]
[tex]\[ h(3) = -(3^2) = -9 \][/tex]
Clearly:
[tex]\[ g(3) = 9 \][/tex]
[tex]\[ h(3) = -9 \][/tex]
[tex]\[ 9 \not< -9 \][/tex]
Therefore, this statement is false.
5. For positive values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex].
For positive [tex]\( x \)[/tex]:
[tex]\[ g(x) = x^2 \][/tex]
[tex]\[ h(x) = -x^2 \][/tex]
Since any positive [tex]\( x \)[/tex] squared is still positive:
[tex]\[ x^2 > -x^2 \][/tex]
Therefore, this statement is true.
6. For negative values of [tex]\( x \)[/tex], [tex]\( g(x) > h(x) \)[/tex].
For negative [tex]\( x \)[/tex]:
[tex]\[ g(x) = x^2 \][/tex]
[tex]\[ h(x) = -x^2 \][/tex]
Since any negative [tex]\( x \)[/tex] squared is still positive:
[tex]\[ x^2 > -x^2 \][/tex]
Therefore, this statement is true.
Thus, 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].
