Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x+1/x^5

Verified step by step guidance

1. Step 1: Recall the definitions of even and odd functions. 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 \( f(x) = x + \frac{1}{x^5} \) to find \( f(-x) \). This gives \( f(-x) = -x + \frac{1}{(-x)^5} \).
3. Step 3: Simplify \( f(-x) \). Since \((-x)^5 = -x^5\), we have \( f(-x) = -x - \frac{1}{x^5} \).
4. Step 4: Compare \( f(-x) \) with \( f(x) \). Check if \( f(-x) = f(x) \) to determine if the function is even, or if \( f(-x) = -f(x) \) to determine if the function is odd.
5. Step 5: Conclude whether the function is even, odd, or neither based on the comparison in Step 4.

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. 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 the 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 is f(x) = x + 1, where substituting -x yields a different result that does not fit either symmetry.