Even and Odd Functions


In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.


f(x) = x⁻⁵

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. 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) for all x in the domain.
2. Let's start by finding f(-x) for the given function f(x) = x⁻⁵. Substitute -x into the function: f(-x) = (-x)⁻⁵.
3. Simplify the expression f(-x) = (-x)⁻⁵. Since the exponent is an odd number, we have f(-x) = -x⁻⁵.
4. Now, compare f(-x) with f(x). We have f(-x) = -x⁻⁵ and f(x) = x⁻⁵. Notice that f(-x) = -f(x), which satisfies the condition for the function to be odd.
5. Therefore, the function f(x) = x⁻⁵ is an odd function because f(-x) = -f(x) for all x in its domain.

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) = x² is even because f(-x) = (-x)² = x².

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) = x³, as f(-x) = (-x)³ = -x³.

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 f(-x) does not equal f(x) or -f(x) for all x in its domain. 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).