Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=0.5x^4-2x^2+6

Verified step by step guidance

1. Step 1: Understand the definitions: A function 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 $f(x)=0.5x^4-2x^2+6$ to find $f(-x)$.
3. Step 3: Calculate $f(-x)=0.5(-x{)}^4-2(-x{)}^2+6$. Simplify this expression.
4. Step 4: Compare $f(-x)$ with $f(x)$ to determine if $f(-x)=f(x)$ (even) or $f(-x)=-f(x)$ (odd).
5. Step 5: Conclude whether the function is even, odd, or neither based on the comparison.

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. Common examples include polynomial functions with only even powers of x, such as f(x) = x^2.

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. Typical examples include polynomial functions with only odd powers of x, like f(x) = x^3.

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 can occur when a function contains both even and odd powers of x or has constant terms that disrupt symmetry. An example is f(x) = x^3 + 2, which does not exhibit symmetry about the y-axis or the origin.