Question

Say whether the following function is even, odd, or neither. give reasons for your answer.

h(x) = − 6x³ + 3x² + 5

The function______ even, since it_____ true that______the function____odd, since it___true that_______

Answer 1

The function h(x) = -6x³ + 3x² + 5 is neither even nor odd because it does not satisfy the conditions for even functions (h(x) = h(-x)) or odd functions (h(x) = -h(-x)).

Step-by-step explanation:

The function h(x) = − 6x³ + 3x² + 5 can be determined to be even, odd, or neither by evaluating h(-x) and comparing it to h(x). For an even function, h(x) = h(-x) holds true, and for an odd function, h(x) = -h(-x) holds true.

Let's substitute -x into the function: h(-x) = -6(-x)³ + 3(-x)² + 5 = 6x³ + 3x² + 5. We can observe that h(-x) is not equal to h(x) nor is it equal to -h(x), so h(x) is neither even nor odd.

An even function shows symmetry around the y-axis, and an odd function shows symmetry about the origin. In this case, due to the presence of both an odd term (− 6x³) and even terms (3x² and the constant 5), the function does not exhibit the required symmetry to be classified as either.