Question

F(x)=-x⁴+14x²+275, complete find whether F is even, odd, or neither.

Answer 1

The function F(x) = -x⁴ + 14x² + 275 is an even function because it satisfies the condition of being symmetric about the y-axis: F(x) = F(-x).

Step-by-step explanation:

To determine whether the function F(x) is even, odd, or neither, we need to apply certain symmetry tests to F(x) = -x⁴ + 14x² + 275. An even function satisfies the condition f(x) = f(-x), meaning that the function is symmetric about the y-axis. On the other hand, an odd function satisfies the condition f(-x) = -f(x), which implies that the function is symmetric about the origin.

First, we evaluate F(x):

F(x) = -(x⁴) + 14(x²) + 275

Next, we evaluate F(-x):

F(-x) = -((-x)⁴) + 14((-x)²) + 275

Because (-x)⁴ = x⁴ and (-x)² = x², we realize:

F(-x) = -(x⁴) + 14(x²) + 275

Since F(x) and F(-x) are identical, F(x) is an even function.