Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x^4+4/x^2

Verified step by step guidance

1. Step 1: Understand the definitions: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain. A function is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain.
2. Step 2: Substitute \( -x \) into the function \( f(x) = x^4 + \frac{4}{x^2} \) to find \( f(-x) \). This gives \( f(-x) = (-x)^4 + \frac{4}{(-x)^2} \).
3. Step 3: Simplify \( f(-x) \). Since \((-x)^4 = x^4\) and \((-x)^2 = x^2\), we have \( f(-x) = x^4 + \frac{4}{x^2} \).
4. Step 4: Compare \( f(-x) \) with \( f(x) \). Notice that \( f(-x) = f(x) \), which satisfies the condition for an even function.
5. Step 5: Conclude that the function \( f(x) = x^4 + \frac{4}{x^2} \) is even because \( f(-x) = f(x) \).

Key Concepts

Even Functions

A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of the function is symmetric with respect to the y-axis. For example, the function f(x) = x^2 is even because f(-x) = (-x)^2 = x^2.

Odd Functions

A function is considered odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of the function is symmetric with respect to the origin. An example of an odd function is f(x) = x^3, as f(-x) = (-x)^3 = -x^3.

Neither Even Nor Odd Functions

A function is classified as neither even nor odd if it does not satisfy the conditions for either classification. This means that the function does not exhibit symmetry about the y-axis or the origin. For instance, the function f(x) = x + 1 is neither even nor odd, as f(-x) = -x + 1 does not equal f(x) or -f(x).