Odd Function — Definition, Graph & Examples
Odd Function
A function with a graph that is symmetric with respect to the origin. A function is odd if and only if f(–x) = –f(x).
See also
Key Formula
f(−x)=−f(x)for all x in the domain
Where:
- f = The function being tested for oddness
- x = Any value in the domain of the function
- −x = The opposite (negative) of x
- −f(x) = The opposite (negative) of the function's output at x
Worked Example
Problem: Determine whether the function f(x) = x³ − x is odd, even, or neither.
Step 1: Write out f(−x) by replacing every x in the formula with −x.
f(−x)=(−x)3−(−x)=−x3+x
Step 2: Write out −f(x) by negating the entire original function.
−f(x)=−(x3−x)=−x3+x
Step 3: Compare the two results from Steps 1 and 2.
f(−x)=−x3+x=−f(x)
Step 4: Since f(−x) equals −f(x) for all x, the condition for an odd function is satisfied.
Answer: f(x) = x³ − x is an odd function.
Another Example
This example shows how adding a constant to an otherwise odd function destroys the odd property. It also demonstrates that a function can be neither odd nor even.
Problem: Determine whether the function g(x) = x³ + 2 is odd, even, or neither.
Step 1: Compute g(−x) by substituting −x into the function.
g(−x)=(−x)3+2=−x3+2
Step 2: Compute −g(x) by negating the entire original function.
−g(x)=−(x3+2)=−x3−2
Step 3: Compare the two expressions. Note that −x³ + 2 ≠ −x³ − 2, so g(−x) ≠ −g(x). The function is not odd.
g(−x)=−x3+2=−x3−2=−g(x)
Step 4: Also check for evenness: g(−x) = −x³ + 2 while g(x) = x³ + 2. These are not equal either, so g is not even.
g(−x)=−x3+2=x3+2=g(x)
Answer: g(x) = x³ + 2 is neither odd nor even. The constant term +2 breaks the origin symmetry.
Frequently Asked Questions
How do you tell if a function is odd or even?
Compute f(−x) and compare it to both f(x) and −f(x). If f(−x) = −f(x) for all x, the function is odd. If f(−x) = f(x) for all x, the function is even. If neither identity holds, the function is neither odd nor even. A function can also be both odd and even, but only f(x) = 0 satisfies both conditions.
Does an odd function always pass through the origin?
If 0 is in the domain of an odd function, then yes — f(0) must equal 0. This follows directly from the definition: f(−0) = −f(0) gives f(0) = −f(0), which forces f(0) = 0. However, some odd functions (like f(x) = 1/x) have domains that do not include 0, so the graph does not pass through the origin.
Why is x³ odd but x² even?
When you substitute −x into x³, you get (−x)³ = −x³ = −f(x), which satisfies the odd condition. For x², substituting gives (−x)² = x² = f(x), which satisfies the even condition. In general, power functions xⁿ are odd when n is an odd integer and even when n is an even integer — this is actually where the names come from.
Odd Function vs. Even Function
| Odd Function | Even Function | |
|---|---|---|
| Defining condition | f(−x) = −f(x) | f(−x) = f(x) |
| Symmetry type | Symmetric about the origin (180° rotational symmetry) | Symmetric about the y-axis (mirror symmetry) |
| Value at x = 0 | f(0) = 0 (if 0 is in the domain) | f(0) can be any value |
| Classic examples | x, x³, sin(x), tan(x) | x², x⁴, cos(x), |x| |
| Integral over [−a, a] | Always equals 0 | Equals 2 times the integral over [0, a] |
Why It Matters
Recognizing odd functions saves significant work in calculus: the definite integral of any odd function over a symmetric interval [−a, a] is always zero, so you can skip the computation entirely. In Fourier analysis, odd functions are represented purely by sine terms, which is essential in physics and engineering. Odd and even function decomposition also appears in precalculus and algebra courses whenever you analyze symmetry of graphs.
Common Mistakes
Mistake: Assuming a function is odd just because it contains odd powers of x.
Correction: A constant term or even-powered term mixed in can break the odd symmetry. For example, x³ + 1 is neither odd nor even. Always verify by computing f(−x) and checking whether it equals −f(x) for the entire expression.
Mistake: Forgetting to negate the entire function when computing −f(x).
Correction: When you form −f(x), distribute the negative sign to every term. For f(x) = x³ − 4x, you need −f(x) = −x³ + 4x, not −x³ − 4x. Failing to negate every term will lead to a wrong conclusion.
Related Terms
- Even Function — Counterpart: symmetric about the y-axis
- Function — General concept that odd functions are a type of
- Symmetric with Respect to the Origin — The geometric property defining odd functions
- Graph of an Equation or Inequality — Visual representation used to identify symmetry
- If and Only If — Logical connective used in the definition
- Sine — A fundamental trigonometric odd function
- Polynomial — Odd polynomials contain only odd-degree terms