Which statements are true for the functions g(x) = x² and h(x) = -x² ? check all that apply.
For any value of x, g(x) will always be greater than h(x)
For any value of x, h(x) will always be greater than g(x)
g(x) > h(x) for x = -1
g(x) < h(x) for x = 3
For positive values of x, g(x) > h(x)
For negative values of x, g(x) > h(x)

Solution:

It is given that

Functions g(x) = x² and h(x) = -x².

Consider statement (a), for any value of x, g(x) will always be greater than h(x).

If x = 0 then, they have the same value.

So, as we can see, for any real value of x, the value of g(x) is always positive and the value of h(x) is always negative.

But if the value of x is imaginary then the given statement is False.

Then consider statement (b), for any value of x, h(x) will always be greater than g(x).

We take the imaginary value of x then the given statement may be satisfied.

But for the real values of the x, the given statement is False.

Now consider statement (c), g(x) > h(x) for x = -1,

As per the statement, x = - 1

g(-1) = 1 and h(-1) = -1

So, by above solution it is clear that,

g(x) > h(x)

This statement (c) is True.

Consider statement (d), g(x) < h(x) for x = 3,

As per the statement, x = 3

g(3) = 9 and h(3) = -9

So, by above solution it is clear that,

g(x) > h(x)

This statement (d) is False.

Consider statement (e), For positive values of x, g(x) > h(x),

As per the statement, for positive values of x, g(x) is always positive and h(x) is always negative.

x = 3

g(3) = 9 and h(3) = -9

So, by above solution it is clear that,

g(x) > h(x)

This statement (e) is True.

Consider statement (f), For positive values of x, g(x) > h(x),

As per the statement, for negative values of x, g(x) is always positive and h(x) is always negative.

x = - 1

g(-1) = 1 and h(-1) = -1

So, by above solution it is clear that,

g(x) > h(x)

This statement (f) is True.

Therefore, statements (c), (e), (f) are true for the functions g(x) = x² and h(x) = -x²

---

Which statements are true for the functions g(x) = x² and h(x) = -x² ? check all that apply.
For any value of x, g(x) will always be greater than h(x)
For any value of x, h(x) will always be greater than g(x)
g(x) > h(x) for x = -1
g(x) < h(x) for x = 3
For positive values of x, g(x) > h(x)
For negative values of x, g(x) > h(x)

Summary:

Statements (c), (e), (f) are true for the functions g(x) = x² and h(x) = -x². i.e g(x) > h(x) for x = -1, For positive values of x, g(x) > h(x) and for negative values of x, g(x) > h(x) are the correct statements.