Even and odd functions; symmetry of graphs
Even and Odd Functions
Even and Odd Functions is a topic in Functions in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.2(D), which requires students to describe and analyze the symmetry of even and odd functions.
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.