Exercises 107–109 will help you prepare for the material covered in the next section. Determine whether f(x)=x^4−2x^2+1 is even, odd, or neither. Describe the symmetry, if any, for the graph of f.

Key Concepts

Even and Odd Functions

A function is classified as even if f(-x) = f(x) for all x in its domain, indicating symmetry about the y-axis. Conversely, a function is odd if f(-x) = -f(x), which shows symmetry about the origin. Understanding these definitions is crucial for determining the nature of the function f(x) = x^4 - 2x^2 + 1.

Graphical Symmetry

Graphical symmetry refers to the visual characteristics of a function's graph. An even function will reflect across the y-axis, while an odd function will exhibit rotational symmetry around the origin. Identifying these symmetries helps in predicting the behavior of the function's graph without plotting it entirely.

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. The function f(x) = x^4 - 2x^2 + 1 is a polynomial of degree 4, and its behavior, including symmetry and end behavior, can be analyzed using its degree and leading coefficient.