Even and Odd Functions Algebraically
Description/Explanation/Highlights
Video Description of Even and Odd Functions Algebraically
This video explains how to determine if a function is even, odd, or neither by using Algebra.
Steps and Key Points to Remember When Determining Even and Odd Functions Algebraically
To determine if a function is even, odd, or neither Algebraically, follow these steps:
- If a function is odd then \(f(-x)=-f(x)\).
- If a function is even then \(f(-x)=f(x)\).
- A function that is neither even nor odd is classified as neither.
- To find out if a function is even or odd, first calculate \(f(-x)\). In the example \(f(x)=x^5+2x^3-x\), first find \(f(-x)\) by substituting -x for each x in the original function. \(f(-x)=(-x)^5+2(-x)^3-(-x)\longrightarrow -x^5-2x^3+x\).
- We now know that the function is not even since f(-x) is not the same as the original function. Factor a negative out of f(-x) to get \(-(x^5+2x^3-x)\). Notice that the part left in the parenthesis is the original function meaning that the \(f(-x)=-f(x)\) and therefore the function is odd.
- We will start the same way with the example \(f(x)=x^4+3x^2-2\) and find \(f(-x)\). \(f(-x)=(-x)^4+3(-x)^2-2\longrightarrow x^4+3x^2-2\). Notice that this is equal to the original function so since \(f(-x)=f(x)\), this function is even and we are finished.
- In the example \(f(x)=2x^3-x^2+x\), substituting -x gets \(2(-x)^3-(-x)^2+(-x)\longrightarrow -2x^3-x^2-x\) showing it is not even and then factoring out a negative gives us \(-(2x^3+x^2+x)\) which shows us the function is not odd since the part in the parenthesis is not the same as the original function, therefore the function is neither.
Here are some key points to keep in mind when determining if a function is even, odd, or neither using Algebra:
- If \(f(-x)=-f(x)\) the function is odd.
- If \(f(-x)=f(x)\) the function is even.
- Start by finding the \(f(-x)\) of the original to determine if the function is even.
- If the function is not even, factor out a negative from what you just found for \(f(-x)\) to see if the part without the negative is the same as the original function which would make it odd.
- If neither are true, the function is neither even nor odd.
Video Highlights
- 00:00 Introduction
- 00:14 Algebraic definition of an even or odd function
- 00:35 \(f(x)=x^5+2x^3-x\) example of determining an odd function Algebraically
- 02:42 \(f(x)=x^4+3x^2-2\) example of determining an even function Algebraically
- 04:01 \(f(x)=2x^3-x^2+x\) example of determining that a function is neither even nor odd
- 06:30 Conclusion
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