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.

Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problema 109

Capitolo 4, Problema 109

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.

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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.

Start by finding $ f(-x) $ for the given function $ f(x) = x^{4} - 2x^{2} + 1 $. Substitute $ -x $ into the function:

\[ f(-x) = (-x)^{4} - 2(-x)^{2} + 1 \]

Simplify each term using the properties of exponents:

\[ (-x)^{4} = x^{4} \quad \text{and} \quad (-x)^{2} = x^{2} \]

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.

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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.

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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.

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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.

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