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.