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 \( x \), and it is odd if \( f(-x) = -f(x) \) for all \( x \). 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:
3. \[ f(-x) = (-x)^{4} - 2(-x)^{2} + 1 \]
4. Simplify each term using the properties of exponents:
5. \[ (-x)^{4} = x^{4} \quad \text{and} \quad (-x)^{2} = x^{2} \]
6. So, \( f(-x) = x^{4} - 2x^{2} + 1 \). Compare this with \( f(x) \) to determine if the function is even, odd, or neither, and then describe the symmetry of the graph based on your conclusion.

Key Concepts

Even and Odd Functions

A function 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. Determining whether a function is even, odd, or neither helps describe its graph's symmetry.

Function Substitution and Simplification

To test if a function is even or odd, substitute -x into the function and simplify. Comparing the result to the original function f(x) or its negative -f(x) reveals the function's symmetry properties. Accurate algebraic manipulation is essential for this step.

Graph Symmetry

Graph symmetry relates to the visual properties of the function's graph. Even functions are symmetric about the y-axis, meaning the left and right sides mirror each other. Odd functions have rotational symmetry about the origin, where rotating the graph 180 degrees yields the same graph.