Question

Which statements are true for the functions g(x) = x² and h(x) = -x²?

1) For any value of x, h(x) will always be greater than g(x).
2) g(x) < h(x) for x = 3.
3) For positive values of x, g(x) > h(x).
4) For negative values of x, g(x) > h(x).
5) g(x) > h(x) for x = -1.

Answer 1

The statements 1) and 2) are false, while statements 3), 4), and 5) regarding the functions g(x) = x² and h(x) = -x² are true. For positive and negative values of x, g(x) will always be greater than h(x).

Step-by-step explanation:

To evaluate the statements about the functions g(x) = x² and h(x) = -x², we need to understand how each function behaves for different values of x.

• For any value of x, since g(x) is x squared and h(x) is the negative of x squared, g(x) will always be non-negative while h(x) will always be non-positive. Therefore, the first statement is false.
• At x = 3, g(3) = 3² = 9 and h(3) = -3² = -9. Thus, g(x) is greater than h(x), making the second statement false.
• For positive values of x, g(x), being the square of x, will always be positive, while h(x) will be the negative of that square, hence negative. So, g(x) is greater than h(x), and the third statement is true.
• For negative values of x, g(x) will still be positive (since squaring a negative number results in a positive number), and h(x) will be negative. Thus, g(x) is greater than h(x) for negative x as well, making the fourth statement true.
• At x = -1, g(-1) = (-1)² = 1 and h(-1) = -(-1)² = -1. Hence, g(x) at x = -1 is greater than h(x), confirming that the fifth statement is also true.