Enter function expression to determine even/odd/neither, analyze f(-x) symmetry
Select Function Type
Enter polynomial expression (use ^ for power, e.g., x^2+2*x-1)
Enter radicand expression (function is √(expression))
√()
Enter logarithm argument expression (function is ln|expression|)
ln||
Enter absolute value expression (function is |expression|)
||
Result
Step-by-Step Derivation
Parity Rules
Even: f(-x) = f(x) Graph symmetric about y-axis
Odd: f(-x) = -f(x) Graph symmetric about origin
Neither: f(-x) ≠ f(x) and f(-x) ≠ -f(x)
Function parity reflects graph symmetry properties, with important applications in integral calculus and series expansion.
⚠Polynomials: even-exponent terms are even-function terms, odd-exponent terms are odd-function terms. Note: ln|x| is even (since |-x|=|x|), while ln x is neither even nor odd.
What Is Function Parity?
Function parity describes the symmetry of a function's graph. By analyzing the relationship between f(-x) and f(x), functions are classified as even, odd, or neither.
Even Functions
Satisfy f(-x)=f(x), symmetric about y-axis. Examples: x², x⁴, cos x, |x|.
Odd Functions
Satisfy f(-x)=-f(x), symmetric about origin. Examples: x³, x⁵, sin x, tan x.
Determination Method
Replace all x with -x, simplify, and compare: equal to original = even; equal to opposite = odd; otherwise neither.
Geometric Meaning
Parity has important applications in integration: odd functions integrate to zero over symmetric intervals, even integrals can be simplified to 2x half-interval.
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