Question

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

Answer 1

The statements that are true for the functions g(x) = x² and h(x) = -x² are:

1. For any value of x, g(x) will always be greater than h(x).

3. g(x) > h(x) for x = -1.

5. For negative values of x, g(x) > h(x)

The correct statements for the functions g(x) = x² and h(x) = -x² are as follows:

1. For any value of x, g(x) will always be greater than h(x).

- Example: When x = 2, g(x) = 2² = 4, and h(x) = -(2²) = -4. Therefore, g(x) > h(x).

3. g(x) > h(x) for x = -1.

- Example: When x = -1, g(x) = (-1)² = 1, and h(x) = -(-1)² = -1. Therefore, g(x) > h(x).

5. For negative values of x, g(x) > h(x).

- Example: When x = -2, g(x) = (-2)² = 4, and h(x) = -(-2)² = -4. Therefore, g(x) > h(x).

Answer 2

A, C, E & F are true

Explanation:

We are given the functions;

g(x) = x² and h(x)= -x²

Let's analyze the options;

A) For any value of x,g(x) will always be greater than h(x);

Let's put x = 1 in both functions;

g(x) = 1² = 1

h(x) = - (1)² = - 1

Let's try a negative number say -1:

g(x) = (-1)² = 1

h(x) = -(-1)² = -1

In both cases, we see that g(x) > h(x) and so the statement is true.

B) For any value of x,h(x) will always be greater than g(x); As seen in A above, for any value of x, h(x) will always be greater than g(x). And so this statement here is wrong.

C) g(x) > h(x) for x = -1;

As seen in A above, at x = -1, g(x) > h(x).

Thus, this statement is true

D) g(x) < h(x) for x = 3;

g(x) = (3)² = 9

h(x) = -(3)² = -9

g(x) is not less than h(x) and so the statement is not correct.

E) For positive values of x, g(x) > h(x).

We have tried positive numbers at x = 1 and x = 3 in previous answers above and in both cases, g(x) > h(x).

Thus, statement is true.

F) For negative values of x, g(x) > h(x).

We have seen earlier that at a negative value of x = -1, g(x) > h(x)

Let's try x = -2

g(x) = (-2)² = 4

h(x) = -(-2)² = -4

Still seeing that g(x) > h(x).

Thus the statement is true.