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

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

All textbooks

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 109

Chapter 4, Problem 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.

Verified step by step guidance

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Recommended video:

Guided course