Exercises 107–109 will help you prepare for the material covered in the next section. Determine whether f(x)=x⁴−2x²+1 is even, odd, or neither. Describe the symmetry, if any, for the graph of f.

Verified step by step guidance

1. Recall the definitions: A function f(x) is even if f(-x) = f(x) for all x in the domain, and it is odd if f(-x) = -f(x) for all x in the domain. If neither condition holds, the function is neither even nor odd.
2. Start by finding f(-x) for the given function f(x) = x^4 - 2x^2 + 1. Substitute -x into the function: f(-x) = (-x)^4 - 2(-x)^2 + 1.
3. Simplify the expression for f(-x): Since (-x)^4 = x^4 and (-x)^2 = x^2, this becomes f(-x) = x^4 - 2x^2 + 1.
4. Compare f(-x) with f(x): Since f(-x) = x^4 - 2x^2 + 1 and f(x) = x^4 - 2x^2 + 1, they are equal, so f(-x) = f(x).
5. Conclude that the function f(x) is even, which means its graph is symmetric with respect to the y-axis.

Key Concepts

Even and Odd Functions

A function f(x) is even if f(-x) = f(x) for all x in its domain, indicating symmetry about the y-axis. It is odd if f(-x) = -f(x), showing symmetry about the origin. If neither condition holds, the function is neither even nor odd.

Function Symmetry

Symmetry in functions relates to how their graphs reflect across axes or points. Even functions have y-axis symmetry, meaning the left and right sides of the graph mirror each other. Odd functions have origin symmetry, where rotating the graph 180 degrees about the origin leaves it unchanged.

Evaluating Function Expressions

To determine if a function is even or odd, substitute -x into the function and simplify. Comparing f(-x) to f(x) and -f(x) helps identify symmetry. This process requires careful algebraic manipulation of polynomial expressions.