Even and odd functions describe special kinds of symmetry in graphs and formulas. They are useful because symmetry can make graphing faster, simplify calculations, and reveal patterns in equations. An even function matches itself when x is replaced by -x, while an odd function changes sign when x is replaced by -x.
These ideas appear often in algebra, trigonometry, calculus, and physics.
Key Facts
- Even function test: f(-x) = f(x) for every x in the domain.
- Odd function test: f(-x) = -f(x) for every x in the domain.
- Even functions have symmetry across the y-axis.
- Odd functions have rotational symmetry of 180 degrees about the origin.
- Example of an even function: f(x) = x^2 because f(-x) = (-x)^2 = x^2.
- Example of an odd function: f(x) = x^3 because f(-x) = (-x)^3 = -x^3.
Vocabulary
- Even function
- A function is even if replacing x with -x gives the same output, so f(-x) = f(x).
- Odd function
- A function is odd if replacing x with -x gives the opposite output, so f(-x) = -f(x).
- Y-axis symmetry
- Y-axis symmetry means the left and right sides of a graph are mirror images across the y-axis.
- Origin symmetry
- Origin symmetry means a graph looks the same after a 180 degree rotation around the origin.
- Domain
- The domain of a function is the set of all input values for which the function is defined.
Common Mistakes to Avoid
- Testing only one value of x, because a function must satisfy the even or odd test for every x in its domain, not just for one example.
- Thinking every function is either even or odd, because many functions are neither, such as f(x) = x^2 + x.
- Confusing y-axis symmetry with origin symmetry, because y-axis symmetry means f(-x) = f(x), while origin symmetry means f(-x) = -f(x).
- Forgetting to check the domain, because even and odd symmetry requires that if x is in the domain, then -x must also be in the domain.
Practice Questions
- 1 Determine whether f(x) = 4x^2 - 7 is even, odd, or neither by computing f(-x).
- 2 Determine whether g(x) = 3x^5 - 2x is even, odd, or neither by computing g(-x).
- 3 A graph has mirror symmetry across the y-axis but does not pass through the origin. Explain whether the function is even, odd, both, or neither.