Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x^4-5x+8

Verified step by step guidance

1. To determine if a function is even, odd, or neither, we need to analyze the function's symmetry properties.
2. First, recall that a function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \). This means the function is symmetric with respect to the y-axis.
3. A function is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain of \( f \). This means the function is symmetric with respect to the origin.
4. Let's substitute \( -x \) into the function \( f(x) = x^4 - 5x + 8 \) to find \( f(-x) \). Calculate \( f(-x) = (-x)^4 - 5(-x) + 8 \).
5. Simplify \( f(-x) \) to see if it equals \( f(x) \) or \(-f(x)\). If neither condition is met, the function is neither even nor odd.

Key Concepts

Even Functions

A function is considered 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 classified as 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 neither even nor odd if it does not satisfy the conditions for either classification. This means that the function's graph lacks symmetry with respect to both the y-axis and the origin. For instance, the function f(x) = x^2 + x is neither even nor odd, as it does not fulfill the criteria for either symmetry.