Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

3. Functions

Intro to Functions & Their Graphs

College Algebra

3. Functions

Intro to Functions & Their Graphs

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1:17 minutes

Problem 59

Textbook Question

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=0.5x^4-2x^2+6

Verified step by step guidance

Step 1: Understand the definitions: A function is even if \( f(-x) = f(x) \) for all \( x \) in the domain. A function is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain.

Step 2: Substitute \( -x \) into the function \( f(x) = 0.5x^4 - 2x^2 + 6 \) to find \( f(-x) \).

Step 3: Calculate \( f(-x) = 0.5(-x)^4 - 2(-x)^2 + 6 \). Simplify this expression.

Step 4: Compare \( f(-x) \) with \( f(x) \) to determine if \( f(-x) = f(x) \) (even) or \( f(-x) = -f(x) \) (odd).

Step 5: Conclude whether the function is even, odd, or neither based on the comparison.

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Key Concepts

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

Even Functions

A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of the function is symmetric with respect to the y-axis. Common examples include polynomial functions with only even powers of x, such as f(x) = x^2.

Odd Functions

A function is considered odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of the function is symmetric with respect to the origin. Typical examples include polynomial functions with only odd powers of x, like f(x) = x^3.

Neither Even Nor Odd Functions

A function is classified as neither even nor odd if it does not satisfy the conditions for either classification. This can occur when a function contains both even and odd powers of x or has constant terms that disrupt symmetry. An example is f(x) = x^3 + 2, which does not exhibit symmetry about the y-axis or the origin.