Question

Determine whether the function f (x) = 9x^3 - 4 + 4x^4
is even, odd or neither.

Answer 1

To determine whether a function is even, odd, or neither, we can check its symmetry.

1. Even function: A function f(x) is even if f(x) = f(-x) for every x in the domain. This means that the graph of the function is symmetric with respect to the y-axis.

2. Odd function: A function f(x) is odd if f(-x) = -f(x) for every x in the domain. This means that the graph of the function is symmetric with respect to the origin.

Let's check the function f(x) = 9x^3 - 4 + 4x^4:

1. Checking for even symmetry:
f(x) = f(-x)
9x^3 - 4 + 4x^4 = 9(-x)^3 - 4 + 4(-x)^4
This simplifies to:
9x^3 - 4 + 4x^4 = -9x^3 - 4 + 4x^4
The equation holds true, so the function is even.

2. Checking for odd symmetry:
f(-x) = -f(x)
9(-x)^3 - 4 + 4(-x)^4 = -9x^3 + 4 - 4x^4
This simplifies to:
-9x^3 - 4 + 4x^4 = -9x^3 + 4 - 4x^4
The equation does not hold true, so the function is not odd.

Therefore, the function f(x) = 9x^3 - 4 + 4x^4 is even.