Test a function
Use x, operators, parentheses, and supported math functions.
Example data table
Use these familiar functions to check the classification rules.
| Function | f(−x) | Classification | Reason |
|---|---|---|---|
| x² + 4 | (−x)² + 4 = f(x) | Even | It reflects across the y-axis. |
| x³ − 2x | −x³ + 2x = −f(x) | Odd | It rotates 180 degrees about the origin. |
| x² + x | x² − x | Neither | It matches neither rule for general x. |
| 0 | 0 | Both | Zero satisfies both equalities. |
Formula used
The calculator evaluates both expressions across many points. It allows small floating-point differences using the selected tolerance. A function is marked neither when at least one valid point fails each rule.
How to use this calculator
- Enter f(x) with x as the variable. Use
^for powers. - Choose a domain that includes matching positive and negative values.
- Set sample points and tolerance. More samples give a stronger numeric check.
- Select degrees only for degree-based trigonometric expressions.
- Press Classify Function. Review the result, graph, table, CSV, or PDF export.
Understanding function symmetry
Even symmetry
Function symmetry gives a fast way to understand a graph. An even function has y-axis symmetry. Its left side mirrors its right side. Replacing x with negative x leaves the output unchanged. For example, x² and cos(x) are even. Their signs may change inside the formula. The final output remains identical.
Odd symmetry
An odd function has origin symmetry. Rotate its graph 180 degrees around the origin. The graph lands on itself. Replacing x with negative x reverses the final output. The rule is f(−x) = −f(x). Common examples include x³, x⁵, and sin(x). Odd functions usually pass through the origin. This applies when zero belongs to the domain.
When no rule fits
Many functions are neither even nor odd. A mixed expression often creates this result. For example, x² + x includes an even term and an odd term. The terms do not combine into one symmetry rule. A constant other than zero also breaks odd symmetry. A horizontal shift usually breaks simple symmetry too.
Domain and numeric testing
Start by checking the domain. The domain must be symmetric around zero. A function needs matching x and −x inputs. Otherwise, it cannot be even or odd. Rational functions need extra care. Values causing division by zero are invalid. Square roots and logarithms can have limited domains. This calculator reports skipped points for undefined expressions.
The numeric method compares many paired values. It calculates f(x), f(−x), and −f(x). Then it measures each difference. Floating-point calculations can produce tiny rounding errors. The tolerance setting prevents harmless decimals changing the classification. Use a smaller tolerance for strict checks. Use more sample points for broader evidence.
Graphs and symbolic checks
Graphs make the rules easier to see. In an even graph, matching inputs share one height. In an odd graph, points form opposite pairs. One point reflects through the origin. The table also shows these paired outputs. Use the CSV file for homework records. Use the PDF file to save a clean summary.
Numeric testing is helpful, but it is not proof. A symbolic substitution remains the strongest method. Simplify f(−x) carefully. Compare it with f(x) and −f(x). The calculator supports reasoning with data and graphs. It reveals patterns manual substitution may hide. This is especially useful for long expressions and trigonometric terms. That supports careful interpretation.
Frequently asked questions
1. What makes a function even?
A function is even when f(−x) equals f(x) for every x in a symmetric domain. Its graph has y-axis symmetry.
2. What makes a function odd?
A function is odd when f(−x) equals −f(x) for every x in a symmetric domain. Its graph has origin symmetry.
3. Can a function be both even and odd?
Yes. The zero function is both even and odd because 0 equals both 0 and −0. On a symmetric domain, it is the standard example.
4. Is x² + x even or odd?
It is neither. Substitution gives x² − x, which is neither the original expression nor its negative for general x.
5. Why does the calculator use a tolerance?
Computers store many decimals approximately. Tolerance treats very small rounding differences as equal, preventing false failures during numeric comparison.
6. Which functions can I enter?
You can use x, numbers, parentheses, +, −, *, /, ^, sin, cos, tan, sqrt, abs, log, exp, and several standard functions.
7. Why are some points skipped?
A point is skipped when the expression is undefined there. Common causes include division by zero, invalid logarithms, and square roots of negative values.
8. Does an even function always contain x²?
No. It may include any expression that produces the same output for x and −x. Cosine and absolute value are common examples.
9. Does an odd function always pass through the origin?
When zero is in its domain, yes. The odd rule forces f(0) to equal −f(0), so f(0) must be zero.
10. Can a shifted graph be even or odd?
Usually not around the original axes. A horizontal or vertical shift often breaks y-axis or origin symmetry, although another symmetry center may exist.
11. Is numeric classification a proof?
No. It is strong evidence across sampled values. For a formal proof, simplify f(−x) symbolically and compare it with f(x) and −f(x).