Determine whether each function is even, odd, or neither. ƒ(x)=x⁴+4/x²

Verified step by step guidance

1. Recall the definitions: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain, and it is odd if \( f(-x) = -f(x) \). If neither condition holds, the function is neither even nor odd.
2. Start by writing the given function: \( f(x) = x^{4} + \frac{4}{x^{2}} \).
3. Find \( f(-x) \) by substituting \( -x \) into the function: \( f(-x) = (-x)^{4} + \frac{4}{(-x)^{2}} \).
4. Simplify \( f(-x) \): since \( (-x)^{4} = x^{4} \) and \( (-x)^{2} = x^{2} \), this becomes \( f(-x) = x^{4} + \frac{4}{x^{2}} \).
5. Compare \( f(-x) \) with \( f(x) \): since they are equal, conclude that the function \( f(x) = x^{4} + \frac{4}{x^{2}} \) is an even function.

Key Concepts

Even and Odd Functions

An even function satisfies f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. An odd function satisfies f(-x) = -f(x), showing symmetry about the origin. Determining whether a function is even, odd, or neither involves testing these conditions.

Function Substitution

To test if a function is even or odd, substitute -x into the function in place of x. Simplify the resulting expression and compare it to the original function f(x) and its negative -f(x). This substitution is essential for verifying symmetry properties.

Domain Considerations

When analyzing evenness or oddness, ensure that both x and -x are within the function's domain. If the domain is restricted, the function may not be classified as even or odd over its entire domain. For example, division by zero must be avoided when substituting -x.