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Even or Odd Function Calculator

Enter the function in the respective field and select the variable. The calculator will quickly determine whether it is odd, even, or neither.

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Even or Odd Function Calculator

This online Even or Odd Function Calculator helps you quickly determine whether a given function is even, odd, or neither. Because of symmetry properties, evaluating only half the domain is often sufficient. Below, we explain definitions, properties, and examples to better understand even and odd functions.

Definitions: Even, Odd, or Neither Function

Even Function

A function f(x) is even if:

f(-x) = f(x) for all x in the domain. Graphically, even functions are symmetric about the y-axis.

Example: f(x) = x²

Even Function

Properties of Even Functions:

Sum or difference of even functions is even.
Product or quotient of even functions is even.
Composition of two even functions is even.
Composition of an even and an odd function is even.

Odd Function

A function f(x) is odd if:

f(-x) = -f(x) for all x in the domain. Graphically, odd functions are symmetric about the origin.

Example: f(x) = x³

Odd Function

Properties of Odd Functions:

Sum or difference of odd functions is odd.
Product or quotient of odd functions is even.
Composition of two odd functions is odd.
Composition of an odd and an even function is even.

Neither Even Nor Odd

If a function shows no symmetry about the y-axis or origin, it is neither even nor odd.

Example: f(x) = x³ + 1

Neither Even Nor Odd

How to Determine if a Function is Even, Odd, or Neither

1. Check for Even Function

Substitute -x into the function. If f(-x) = f(x), the function is even.

Example: f(x) = 2x² – 3
f(-x) = 2(-x)² – 3
       = 2x² – 3
f(-x) = f(x) → Even

2. Check for Odd Function

Substitute -x into the function. If f(-x) = -f(x), the function is odd.

Example: f(x) = -x⁷ + 8x⁵ – x³ + 6x
f(-x) = -(-x)⁷ + 8(-x)⁵ – (-x)³ + 6(-x)
       = x⁷ - 8x⁵ + x³ - 6x
f(-x) = -f(x) → Odd

3. Check for Neither

If neither condition holds, the function is neither even nor odd.

Example: f(x) = x³ – x² – 1
f(-x) = -x³ – x² – 1
Neither f(-x) = f(x) nor f(-x) = -f(x) → Neither

Set Representation of Odd and Even Numbers

Odd numbers: {2x + 1 : x ∈ Z}

Even numbers: {2x : x ∈ Z}

Non-integers (e.g., 3.462, 7/9) are neither odd nor even.

Using the Calculator

Input:

Enter the function and specify the variable.
Click "Calculate".

Output:

The calculator identifies the function as even, odd, or neither.

FAQs

Is cos(x) an odd function?

No, cosine is even, while sine is odd.

Is tan(x) even or odd?

Tangent and sine are odd, cosine is even. Tan(-x) = -tan(x), Cos(-x) = cos(x), Sin(-x) = -sin(x)

Why is zero even?

Zero is a multiple of 2 (0 = 2×0), so it is considered even.

Conclusion

The online Even or Odd Function Calculator makes it easy to determine function symmetry, which is helpful for graphing, algebraic analysis, and homework verification.

References

Wikipedia: Even and Odd Functions
Lumen Learning: Determine whether a function is even, odd, or neither
Libre Text: Odd and Even Functions

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