Determine whether each function is even, odd, or neither. ƒ(x)=x⁴-5x+8

Key Concepts

Even Functions

An even function satisfies the condition f(-x) = f(x) for all x in its domain. Graphically, even functions are symmetric about the y-axis. Common examples include functions with even powers of x, like x² or x⁴.

Odd Functions

An odd function satisfies the condition f(-x) = -f(x) for all x in its domain. These functions have rotational symmetry about the origin. Examples include functions with odd powers of x, such as x³ or x.

Testing Function Symmetry Algebraically

To determine if a function is even, odd, or neither, substitute -x into the function and simplify. Compare the result to f(x) and -f(x). If equal to f(x), the function is even; if equal to -f(x), it is odd; otherwise, it is neither.