Symmetry: Even Functions, Odd Functions, and the Algebraic Test

Symmetry is a shortcut. If you know a graph is symmetric, plotting the right-hand side gives you the left-hand side free — and the algebraic test that detects it takes about ten seconds.

Even Functions: A Mirror in the y-Axis

A function is even when

Inputs and produce the same output, so the graph is unchanged by a reflection across the -axis.

Familiar even functions: , , , any constant, and .

Odd Functions: A Half-Turn About the Origin

A function is odd when

Feeding in gives the negative of the original output, so the point is matched by — diagonally opposite through the origin. That is exactly the rotation from the reflections lesson, and an odd function is one that survives it unchanged.

Familiar odd functions: , , , , and .

One useful consequence: if an odd function is defined at 0, it must pass through the origin. Setting in gives , which forces .

Where the Names Come From

The names are not arbitrary — they come from the exponents on power functions.

For a polynomial the rule is quick: all even powers → even; all odd powers → odd; a mix → neither. A constant term counts as an even power, since , which is why is neither despite the odd-looking .

How to Tell if a Function Is Even or Odd

1. Replace every with .
2. Simplify carefully, watching the powers.
3. Compare with the original:
   • identical → even
   • the whole original negated → odd
   • neither → neither

The step that decides most answers is knowing that but . An even power absorbs the minus sign; an odd power keeps it.

Even vs Odd Functions: Most Are Neither

For : and . Even would need ; odd would need . It is neither, and “neither” is a complete answer — not a sign you have made a mistake.

One caution when checking with numbers: a single pair of points is enough to disprove symmetry but never enough to prove it. A function can satisfy the odd condition at one input by coincidence and fail it elsewhere. To confirm symmetry, do the algebra.

Symmetry About the x-Axis

A graph can be symmetric about the -axis — but then it is not a function.

This symmetry pairs with : the same input with two different outputs. That is precisely what the vertical line test forbids, so is a relation but not a function.

This is why the square root graph is only the top half of a sideways parabola — taking the principal root discards the bottom half to keep it a function.

Worked Example A: An Even Function

Is even, odd or neither?

Even. All powers are even (the constant counts as ).

Worked Example B: An Odd Function

Is even, odd or neither?

Odd. All powers are odd.

Worked Example C: Neither

Is even, odd or neither?

That is not , and does not match it either. Neither — the mix of an even and an odd power guarantees it.

Worked Example D: A Rational Function

Is even, odd or neither?

Odd, which matches its graph: the two branches of the reciprocal curve sit diagonally opposite through the origin.

Worked Example E: The Trap

Is odd?

and . These differ in the constant, so it is neither.

The makes it look odd, but the is an even-power term. Adding a constant to an odd function destroys the symmetry, because it lifts the graph off the origin.

Common Mistakes to Avoid

• Testing one pair of points and concluding it is symmetric. One pair can disprove, never prove.
• Forgetting that . Even powers absorb the minus sign.
• Treating a constant as an odd-power term. , an even power.
• Assuming every function is one or the other. Most are neither.
• Calling a graph even because it has a mirror line. The mirror must be the -axis specifically.
• Thinking -axis symmetry is a third type for functions. It disqualifies the graph from being a function.
• Negating only the first term. negates the whole expression.

Where Symmetry Leads Next

• Halving the work. Establish symmetry, plot one side, mirror it.
• Reflections. Symmetry is exactly a reflection that changes nothing.
• One-to-one functions. An even function is never one-to-one, since gives two inputs one output — so it has no inverse without restriction.
• Integration. Over a symmetric interval an odd function integrates to zero, and an even one to twice the half.
• Fourier series. Even functions decompose into cosines and odd ones into sines, which is where the terminology pays off.

Practice Problems

Work each one before opening the answer.

Problem 1. Is even, odd or neither?

Problem 2. Is even, odd or neither?

Problem 3. Is even, odd or neither?

Problem 4. Is even, odd or neither?

Problem 5. Is even, odd or neither?

Problem 6. Is even, odd or neither?

Problem 7. Is even, odd or neither?

Problem 8. Is even, odd or neither?

Problem 9. Is even, odd or neither?

Problem 10. Which function is both even and odd?

Problem 11. If is odd and defined at 0, what is ?

Problem 12. Can the graph of a function be symmetric about the -axis?

Quick Reference

Symmetry is a reflection that leaves a graph unchanged, which makes it part of the function transformations picture. Parabolas centred on the -axis are the standard even example and cubics the standard odd one, and an even function can never be one-to-one. More Algebra lessons are available.