Even and Odd Functions: Symmetry of Graphs

Quick answer

A function is even when f(−x) = f(x) for every x in its domain, and its graph is symmetric about the y-axis, like the graph of y = x². It is odd when f(−x) = −f(x), and its graph is symmetric about the origin, like the graph of y = x³. To test a function, replace x with −x and simplify. For polynomials the exponents decide: only even powers make an even function, only odd powers make an odd one, and a mix makes neither.

What you'll learn

Test a function for even or odd symmetry by computing f(−x)
Recognize even and odd symmetry in a graph
Predict the symmetry of a polynomial from its exponents
Use symmetry to fill in values and complete a graph

Two kinds of symmetry

For $f(x) = x^2$ , the inputs $3$ and $-3$ give the same output, $9$ . Every pair of opposite inputs behaves that way, so the graph is its own mirror image in the $y$ -axis. For $g(x) = x^3$ , opposite inputs give opposite outputs: $g(-2) = -8 = -g(2)$ . Its graph looks the same after a half turn about the origin.

A function is even if $f(-x) = f(x)$ for every $x$ in its domain. A function is odd if $f(-x) = -f(x)$ for every $x$ in its domain.

y = x⁴ − 3x²

An even function: y = x⁴ − 3x²

y = x³ − 3x

An odd function: y = x³ − 3x

Why the exponents decide

Replacing $x$ with $-x$ in a power gives

(-x)^n = \begin{cases} x^n & n \text{ even} \\ -x^n & n \text{ odd} \end{cases}

because the minus signs pair off in an even power and one is left over in an odd power. A polynomial with only even powers therefore gets every term back unchanged, and one with only odd powers has every term flip sign. A constant counts as an even power, since $c = cx^0$ . Replacing $x$ with $-x$ leaves even powers alone and flips the sign of odd powers, so the exponents decide the symmetry of a polynomial. That is where the names even and odd come from.

Beyond polynomials

The same test works for any function whose domain is symmetric about $0$ .

$|x|$ is even, because $|-x| = |x|$ .
$\tfrac{1}{x}$ and $\sqrt[3]{x}$ are odd.
$\sqrt{x}$ is neither: its domain $x \ge 0$ has no negative inputs to compare.

An odd function defined at $0$ must pass through the origin, because $f(0) = -f(0)$ forces $f(0) = 0$ .

Worked examples

Common mistakes

Practice problems

Is $f(x) = x^6 - 2x^2$ even, odd or neither?

Answer

Even

Full solution

Both powers are even, so $f(-x) = x^6 - 2x^2 = f(x)$ .
Is $f(x) = 5x^3 - x$ even, odd or neither?

Answer

Odd

Full solution

$f(-x) = -5x^3 + x = -f(x)$ .
Is $f(x) = x^2 + x$ even, odd or neither?

Answer

Neither

Full solution

$f(-x) = x^2 - x$ , which is neither $f(x)$ nor $-f(x) = -x^2 - x$ . At $x = 1$ : $f(1) = 2$ and $f(-1) = 0$ .
Is the constant function $f(x) = 7$ even, odd or neither?

Answer

Even

Full solution

$f(-x) = 7 = f(x)$ . Its graph, a horizontal line, is symmetric about the $y$ -axis.
Is $f(x) = |x| + x^2$ even, odd or neither?

Answer

Even

Full solution

$|-x| = |x|$ and $(-x)^2 = x^2$ , so $f(-x) = f(x)$ .
Is $\displaystyle f(x) = \frac{x}{x^2 - 4}$ even, odd or neither?

Answer

Odd

Full solution

The domain, all $x$ except $\pm 2$ , is symmetric about $0$ , and $f(-x) = \tfrac{-x}{x^2 - 4} = -f(x)$ .
$f$ is odd and $f(3) = -4$ . Find $f(-3)$ and $f(0)$ .

Answer

$f(-3) = 4$ and $f(0) = 0$

Full solution

$f(-3) = -f(3) = 4$ . And $f(0) = -f(0)$ forces $f(0) = 0$ .
The graph of an even function passes through $(-2, 5)$ . Name another point on it.

Answer

$(2, 5)$

Full solution

An even function’s graph is its own mirror image in the $y$ -axis.
Is $f(x) = x^5 + x^3 + 2$ odd?

Answer

No; it is neither even nor odd.

Full solution

The constant $2$ is an even power, and $f(0) = 2 \ne 0$ , which no odd function allows. And $f(-1) = 0$ while $f(1) = 4$ , so it is not even either.
A student finds $f(2) = 0$ and $f(-2) = 0$ for $f(x) = x^3 - 4x$ , and concludes that $f$ is even. What went wrong?

Hint

Compare $f(1)$ and $f(-1)$ .

Answer

One pair of equal values proves nothing. In fact $f$ is odd.

Full solution

$f(1) = -3$ and $f(-1) = 3$ , so $f$ is not even. Algebraically, $f(-x) = -x^3 + 4x = -f(x)$ . At $x = 2$ both $f(2)$ and $-f(2)$ equal $0$ , which is why that one test could not tell the difference.

Frequently asked questions

How do you tell if a function is even or odd?

Compute f(−x) and simplify. If it equals f(x), the function is even. If it equals −f(x), the function is odd. If it is neither, the function is neither.

What does the graph of an even function look like?

It is its own mirror image in the y-axis. The point (a, b) is on the graph exactly when (−a, b) is.

What does the graph of an odd function look like?

It is unchanged by a half turn about the origin. The point (a, b) is on the graph exactly when (−a, −b) is.

Can a function be neither even nor odd?

Yes, most functions are neither. x² + x, for example, fails both tests.

Are sine and cosine even or odd?

Cosine is even, since cos(−x) = cos x. Sine is odd, since sin(−x) = −sin x.

What to learn next

Polynomial Identities and the Binomial TheoremWhat makes an equation an identity, how to prove one, how identities generate Pythagorean triples and mental-math shortcuts, and how Pascal's triangle expands (x + y)ⁿ.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.HSF.BF.B.3Building FunctionsIdentify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
CCSS.MATH.CONTENT.HSF.IF.B.4Interpreting FunctionsFor a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

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