Even Function — Definition, Graph & Examples
Even Function
A function with a graph that is symmetric with respect to the y-axis. A function is even if and only if f(–x) = f(x).
See also
Key Formula
f(−x)=f(x)for all x in the domain of f
Where:
- f = The function being tested for evenness
- x = Any input value in the domain of f
- −x = The negative (opposite) of the input value
Worked Example
Problem: Determine whether f(x) = x⁴ − 3x² + 2 is an even function.
Step 1: Write the defining condition. A function is even if f(−x) = f(x) for all x.
Step 2: Compute f(−x) by replacing every x with −x.
f(−x)=(−x)4−3(−x)2+2
Step 3: Simplify each term. Remember that (−x)⁴ = x⁴ and (−x)² = x².
f(−x)=x4−3x2+2
Step 4: Compare f(−x) with the original f(x).
f(−x)=x4−3x2+2=f(x)
Answer: Since f(−x) = f(x), the function f(x) = x⁴ − 3x² + 2 is even.
Another Example
This example shows a function that fails both the even and odd tests, reinforcing that most functions are neither even nor odd.
Problem: Determine whether g(x) = x³ + x² is even, odd, or neither.
Step 1: Compute g(−x) by substituting −x for x.
g(−x)=(−x)3+(−x)2=−x3+x2
Step 2: Compare g(−x) with g(x). The original is g(x) = x³ + x².
g(−x)=−x3+x2=x3+x2=g(x)
Step 3: Since g(−x) ≠ g(x), the function is not even. Now check if it is odd by testing whether g(−x) = −g(x).
−g(x)=−(x3+x2)=−x3−x2
Step 4: Compare g(−x) with −g(x).
g(−x)=−x3+x2=−x3−x2=−g(x)
Answer: g(x) = x³ + x² is neither even nor odd, because g(−x) equals neither g(x) nor −g(x).
Frequently Asked Questions
What is the difference between an even function and an odd function?
An even function satisfies f(−x) = f(x) and is symmetric about the y-axis. An odd function satisfies f(−x) = −f(x) and is symmetric about the origin. A function can be neither even nor odd, and the only function that is both even and odd is f(x) = 0.
How do you tell if a function is even from its graph?
If you can fold the graph along the y-axis and the two halves match exactly, the function is even. Equivalently, for every point (a, b) on the graph, the point (−a, b) is also on the graph.
Are all polynomial functions with only even exponents even functions?
Yes. A polynomial that contains only even powers of x (including the constant term, which is x⁰) is always an even function. For example, f(x) = 5x⁶ − 2x² + 7 is even because every exponent (6, 2, 0) is even, so replacing x with −x changes nothing.
Even Function vs. Odd Function
| Even Function | Odd Function | |
|---|---|---|
| Algebraic test | f(−x) = f(x) | f(−x) = −f(x) |
| Symmetry | Symmetric about the y-axis | Symmetric about the origin (180° rotation) |
| Value at x = 0 | f(0) can be any value | f(0) must equal 0 (if 0 is in the domain) |
| Common examples | x², x⁴, cos x, |x| | x, x³, x⁵, sin x |
| Graph behavior | Left half mirrors right half across the y-axis | Left half mirrors right half across the origin |
Why It Matters
Recognizing even functions simplifies many problems in algebra and calculus. When you integrate an even function over a symmetric interval [−a, a], you can compute the integral from 0 to a and simply double it. In physics and engineering, identifying symmetry in functions helps reduce the complexity of Fourier series, since even functions have only cosine terms.
Common Mistakes
Mistake: Thinking a function is even just because its exponents happen to be even numbers in one term, while ignoring odd-power terms.
Correction: You must substitute −x for x in the entire function and simplify. If any term changes sign and does not cancel, the function is not even. For example, f(x) = x⁴ + x has an even exponent in the first term but an odd exponent in the second, so f(−x) = x⁴ − x ≠ f(x).
Mistake: Confusing 'even function' with 'even number.' Students sometimes assume the name refers only to polynomial exponents.
Correction: The term 'even function' refers to the algebraic property f(−x) = f(x). Non-polynomial functions like f(x) = cos x and f(x) = |x| are also even. Always apply the definition directly rather than relying on exponent patterns.
Related Terms
- Odd Function — Counterpart: satisfies f(−x) = −f(x)
- Function — General concept that even functions are a type of
- Symmetric with Respect to the y-axis — The geometric property that defines even functions
- Graph of an Equation or Inequality — Visual tool for identifying y-axis symmetry
- If and Only If — Logical connector used in the even-function definition
- Polynomial — Common source of even-function examples
- Cosine — Key trigonometric example of an even function