Symmetry of Functions

Symmetry is a powerful tool for understanding and graphing functions. If you know a function is symmetric, you only need to graph half of it — the other half is determined by the symmetry. In college algebra, we formalize the algebraic tests for symmetry and explore how symmetry simplifies analysis.

Even Functions: Symmetry About the $y$-Axis

A function $f$ is even if:

$f(-x) = f(x) \quad \text{for all } x \text{ in the domain}$

Graphically, an even function is symmetric about the $y$-axis — the left half is a mirror image of the right half. If you fold the graph along the $y$-axis, both sides overlap exactly.

Common even functions:

• $f(x) = x^2$ (and any $x^{2n}$)
• $f(x) = |x|$
• $f(x) = \cos(x)$
• $f(x) = x^4 - 3x^2 + 1$

Algebraic Test for Even Functions

To test whether $f(x) = x^4 - 2x^2 + 5$ is even:

Step 1: Compute $f(-x)$:

$f(-x) = (-x)^4 - 2(-x)^2 + 5 = x^4 - 2x^2 + 5$

Step 2: Compare with $f(x)$:

$f(-x) = x^4 - 2x^2 + 5 = f(x)$

Since $f(-x) = f(x)$, the function is even.

Key insight: A polynomial is even if and only if every term has an even exponent (including the constant, which has exponent 0).

Odd Functions: Symmetry About the Origin

A function $f$ is odd if:

$f(-x) = -f(x) \quad \text{for all } x \text{ in the domain}$

Graphically, an odd function has rotational symmetry about the origin — rotating the graph 180 degrees around the origin produces the same graph. If the point $(a, b)$ is on the graph, then $(-a, -b)$ is also on the graph.

Common odd functions:

• $f(x) = x$ (and any $x^{2n+1}$)
• $f(x) = x^3$
• $f(x) = \sin(x)$
• $f(x) = \frac{1}{x}$

Algebraic Test for Odd Functions

To test whether $g(x) = x^3 - 5x$ is odd:

Step 1: Compute $g(-x)$:

$g(-x) = (-x)^3 - 5(-x) = -x^3 + 5x$

Step 2: Compute $-g(x)$:

$-g(x) = -(x^3 - 5x) = -x^3 + 5x$

Step 3: Compare:

$g(-x) = -x^3 + 5x = -g(x)$

Since $g(-x) = -g(x)$, the function is odd.

Key insight: A polynomial is odd if and only if every term has an odd exponent (and there is no constant term).

Functions That Are Neither Even Nor Odd

Most functions are neither even nor odd. If $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, the function has no symmetry of this type.

Worked Example 1: Testing for Neither

Test $h(x) = x^3 + x^2$ for symmetry.

$h(-x) = (-x)^3 + (-x)^2 = -x^3 + x^2$

Compare with $h(x) = x^3 + x^2$: not equal (the $x^3$ terms differ in sign, but the $x^2$ terms are the same).

Compare with $-h(x) = -x^3 - x^2$: not equal (the $x^2$ terms differ in sign).

Since $h(-x)$ equals neither $h(x)$ nor $-h(x)$, the function is neither even nor odd.

Why? The function mixes even-power terms ($x^2$) with odd-power terms ($x^3$). Symmetry requires all terms to be of one type.

Worked Example 2: Testing a Rational Function

Test $f(x) = \frac{x}{x^2 + 1}$ for symmetry.

$f(-x) = \frac{-x}{(-x)^2 + 1} = \frac{-x}{x^2 + 1} = -\frac{x}{x^2 + 1} = -f(x)$

Since $f(-x) = -f(x)$, the function is odd.

Worked Example 3: Absolute Value with Polynomial

Test $f(x) = |x^3|$ for symmetry.

$f(-x) = |(-x)^3| = |-x^3| = |x^3| = f(x)$

Since $f(-x) = f(x)$, the function is even — even though $x^3$ alone is odd, the absolute value removes the sign difference.

The Special Case: $f(x) = 0$

The zero function $f(x) = 0$ is both even and odd. It is the only function with this property (because it satisfies both $f(-x) = 0 = f(x)$ and $f(-x) = 0 = -f(x)$).

$x$-Axis Symmetry

A graph has $x$-axis symmetry if whenever $(x, y)$ is on the graph, $(x, -y)$ is also on the graph. However, a graph with $x$-axis symmetry cannot be a function (unless it lies entirely on the $x$-axis), because the points $(x, y)$ and $(x, -y)$ represent two outputs for the same input.

Example: The circle $x^2 + y^2 = 25$ has $x$-axis symmetry, $y$-axis symmetry, and origin symmetry — but it is not a function.

Symmetry About Other Lines

A graph can be symmetric about lines other than the axes. The most common case is symmetry about a vertical line $x = h$.

A function $f$ is symmetric about the line $x = h$ if:

$f(h + t) = f(h - t) \quad \text{for all } t$

Example: The parabola $f(x) = (x - 3)^2 + 1$ is symmetric about $x = 3$:

$f(3 + t) = t^2 + 1 = f(3 - t)$

Every quadratic $f(x) = a(x - h)^2 + k$ is symmetric about the line $x = h$ (its axis of symmetry).

Using Symmetry to Simplify Graphing

If you know a function is even, you only need to plot it for $x \geq 0$ and reflect the result across the $y$-axis. If it is odd, plot for $x \geq 0$ and rotate 180 degrees about the origin (equivalently, reflect across $y$-axis then across $x$-axis, or vice versa).

Worked Example 4: Graphing with Symmetry

Graph $f(x) = x^4 - 4x^2$.

Step 1: Test for symmetry.

$f(-x) = (-x)^4 - 4(-x)^2 = x^4 - 4x^2 = f(x)$. The function is even.

Step 2: Find zeros: $x^4 - 4x^2 = x^2(x^2 - 4) = x^2(x - 2)(x + 2) = 0$, giving $x = 0, \pm 2$.

Step 3: Find the minimum. Factor as $f(x) = (x^2 - 2)^2 - 4$. The minimum of $(x^2 - 2)^2$ is 0, occurring when $x^2 = 2$, i.e., $x = \pm\sqrt{2}$. So $f(\pm\sqrt{2}) = -4$.

Step 4: Plot only the right half: $(0, 0)$, $(\sqrt{2}, -4) \approx (1.41, -4)$, $(2, 0)$, and for larger $x$ the function grows rapidly. Mirror across the $y$-axis.

Symmetry and Integration (Calculus Preview)

Even and odd symmetry produce elegant results in calculus:

• The integral of an even function over $[-a, a]$ equals twice the integral over $[0, a]$: $\int_{-a}^{a} f(x)\,dx = 2\int_{0}^{a} f(x)\,dx$

• The integral of an odd function over $[-a, a]$ is always zero: $\int_{-a}^{a} f(x)\,dx = 0$

These properties save significant computation time. While you will not compute integrals in this course, knowing that odd functions “cancel out” over symmetric intervals is conceptually valuable.

Properties of Even and Odd Functions

Several algebraic properties follow from the definitions:

Also, if $f$ is odd, then $f(0) = 0$ (when 0 is in the domain), because $f(-0) = -f(0)$ gives $f(0) = -f(0)$, which means $2f(0) = 0$.

Real-World Application: Structural Engineering

In structural engineering, symmetric loading on a beam produces symmetric deflection curves. If a beam is loaded symmetrically about its midpoint, the deflection function $d(x)$ is even relative to the center:

$d(-x) = d(x)$

Engineers exploit this by analyzing only half the beam. For a fixed-end beam of length $L$ with a uniform load, the deflection relative to the center is:

$d(x) = \frac{w}{24EI}\!\left(x^4 - \frac{L^2}{2}x^2 + \frac{L^4}{16}\right)$

Every term has an even power of $x$, confirming the function is even. The maximum deflection occurs at $x = 0$ (the center).

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A function is even if $f(-x) = f(x)$ — its graph is symmetric about the $y$-axis
• A function is odd if $f(-x) = -f(x)$ — its graph has 180-degree rotational symmetry about the origin
• For polynomials: all even exponents means even; all odd exponents means odd; mixed means neither
• $x$-axis symmetry exists for relations but not for functions (except $f(x) = 0$)
• Symmetry about $x = h$ means $f(h + t) = f(h - t)$ — every parabola has this
• Knowing symmetry lets you graph half the function and derive the rest
• Odd functions always pass through the origin (when 0 is in the domain)
• The product of two odd functions is even; the product of an even and an odd function is odd

Return to College Algebra for more topics in this section.