Question

Is the function H(x) = -x^2 + 5 odd, even, or neither?

A) Odd
B) Even
C) Neither

Answer 1

A function is neither odd nor even if it doesn't satisfy the properties of either. Correct option is C.

Step-by-step explanation:

An even function is symmetric about the y-axis, while an odd function is anti-symmetric about the origin (both the x-axis and y-axis). To determine if a function is odd or even, we need to check if the function satisfies the properties of odd or even functions:

1. Odd function: A function f(x) is odd if f(-x) = -f(x) for all x in the domain. This means that if we replace x with -x, the function value becomes its negative counterpart.
2. Even function: A function f(x) is even if f(-x) = f(x) for all x in the domain. This means that if we replace x with -x, the function value remains the same.

For the function H(x) = -x^2 + 5, let's check both properties:

1. Odd property: H(-x) = -(-x)^2 + 5 = -x^2 + 5
2. Even property: H(-x) = -(-x)^2 + 5 = -x^2 + 5 ≠ H(x)

Since H(x) doesn't satisfy both the odd and even properties, we can conclude that it is neither odd nor even.