Even and odd functions; symmetry of graphs
Even and Odd Functions
Even and Odd Functions is a topic in Functions in the California Common Core State Standards. It is aligned to Standard F-BF.3, which requires students to identify the symmetry of even and odd functions and the effect of transformations.
An even function satisfies \(f(-x)=f(x)\) (symmetric about the \(y\)-axis); an odd function satisfies \(f(-x)=-f(x)\) (symmetric about the origin).
Theory
A function's symmetry can be read straight from what happens when you replace \(x\) with \(-x\).
- Even: \(f(-x)=f(x)\) for every \(x\). The graph is a mirror image across the \(y\)-axis (e.g. \(x^2\), \(\cos x\)).
- Odd: \(f(-x)=-f(x)\) for every \(x\). The graph has 180° rotational symmetry about the origin (e.g. \(x^3\), \(\sin x\)).
- Neither: \(f(-x)\) equals neither \(f(x)\) nor \(-f(x)\).
The two symmetry tests:
A quick shortcut for polynomials: a function of only even powers (and a constant) is even; a function of only odd powers is odd; a mix is usually neither.
How to classify a function
- Compute \(f(-x)\), carefully applying signs: \((-x)^2=x^2\) but \((-x)^3=-x^3\).
- Compare \(f(-x)\) with \(f(x)\): equal \(\Rightarrow\) even.
- Otherwise compare with \(-f(x)\): equal \(\Rightarrow\) odd.
- If neither matches, the function is neither.
Replace \(x\) with \(-x\) and simplify, using \((-x)^4=x^4\) and \((-x)^2=x^2\).
| \(f(-x)\) | \(=\) | \((-x)^4-3(-x)^2\) |
| \(=\) | \(x^4-3x^2\) | |
| \(=\) | \(f(x)\) |
Since \(f(-x)=f(x)\), the function is even (symmetric about the \(y\)-axis).
Compute \(f(-x)\), using \((-x)^3=-x^3\).
| \(f(-x)\) | \(=\) | \((-x)^3-(-x)\) |
| \(=\) | \(-x^3+x\) | |
| \(=\) | \(-(x^3-x)=-f(x)\) |
Since \(f(-x)=-f(x)\), the function is odd (symmetric about the origin).
Find \(f(-x)\) and compare with both \(f(x)\) and \(-f(x)\).
| \(f(-x)\) | \(=\) | \((-x)^2+(-x)=x^2-x\) |
| \(f(x)\) | \(=\) | \(x^2+x\) |
| \(-f(x)\) | \(=\) | \(-x^2-x\) |
\(f(-x)\) matches neither, so \(f\) is neither even nor odd.
Substitute \(-x\); the denominator is unchanged because \((-x)^2=x^2\).
| \(f(-x)\) | \(=\) | \(\dfrac{-x}{(-x)^2+1}\) |
| \(=\) | \(\dfrac{-x}{x^2+1}=-f(x)\) |
So \(f\) is odd.
Common pitfalls
Frequently asked questions
What is an even function?
One where \(f(-x)=f(x)\) for all \(x\); its graph is symmetric about the \(y\)-axis. Examples: \(x^2\), \(\cos x\).
What is an odd function?
One where \(f(-x)=-f(x)\) for all \(x\); its graph has 180-degree rotational symmetry about the origin. Examples: \(x^3\), \(\sin x\).
How do you test whether a function is even or odd?
Substitute \(-x\). If \(f(-x)=f(x)\) it is even; if \(f(-x)=-f(x)\) it is odd; otherwise it is neither.
Can a function be both even and odd?
Only \(f(x)=0\). For any other function the two conditions cannot hold at once.