Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x^5-2x^3

Verified step by step guidance

1. Step 1: Recall the definitions. A function is even if f(-x) = f(x) for all x, and odd if f(-x) = -f(x) for all x.
2. Step 2: Substitute -x for x in the function to find f(-x). For the given function ƒ(x) = x^5 - 2x^3, replace x with -x to get f(-x) = (-x)^5 - 2(-x)^3.
3. Step 3: Simplify f(-x). Remember that raising a negative number to an odd power results in a negative number, so (-x)^5 = -x^5 and (-x)^3 = -x^3. Thus, f(-x) becomes -x^5 + 2x^3.
4. Step 4: Compare f(-x) with f(x) and -f(x). f(x) = x^5 - 2x^3 and -f(x) = -x^5 + 2x^3.
5. Step 5: Determine if the function is even, odd, or neither. Since f(-x) = -f(x), the function ƒ(x) = x^5 - 2x^3 is an odd function.

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 an even function is symmetric with respect to the y-axis. A common example of an even function is f(x) = x^2, where substituting -x yields the same output as substituting x.

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 an odd function is symmetric with respect to the origin. An example of an odd function is f(x) = x^3, where substituting -x results in the negative of the output for x.

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) and also does not equal -f(x). An example would be f(x) = x^2 + x, which does not exhibit symmetry about the y-axis or the origin.