Function Symmetry

5 min read

Reference: Stewart §1.1

Using symmetry

Why compute both $f(5)$ and $f(-5)$ if one tells you the other? Recognizing symmetry in functions can cut your work in half, and it reveals deep structure that matters for integration, series, and beyond.

When you fold a graph along the $y$-axis and the two halves match, that’s an even function. When you rotate a graph 180° around the origin and it looks the same, that’s an odd function. These aren’t just visual curiosities; they’re computational tools.

Prerequisite Map

Quick Reference

Property	Value
Chapter	1 - Functions and Limits
Section	1.1
Difficulty	Intermediate
Time	~20 minutes

Key Concepts

Even and Odd Functions

Type	Algebraic Test	Geometric Symmetry
Even	$f(-x) = f(x)$	Symmetric about the $y$-axis
Odd	$f(-x) = -f(x)$	Symmetric about the origin
Neither	Neither condition holds	No special symmetry

Visual Comparison

    Even: f(-x) = f(x)              Odd: f(-x) = -f(x)

        y                               y
        |                               |
        |    ●       ●                  |    ●
        |   / \     / \                 |   /
        |  /   \   /   \                |  /
        | /     \ /     \               +-/-------→ x
        +--------+-------→ x           /|
                                      / |
        Fold along y-axis            ● |
        and halves match              Rotate 180° around origin
                                      and it looks the same

Testing for Symmetry (Algebraic Method)

Process:

Replace every $x$ with $-x$ in the formula
Simplify the result completely
Compare to the original $f(x)$:
- If result = $f(x)$: Even
- If result = $-f(x)$: Odd
- If neither: Neither

Common Examples

Even Functions	Odd Functions	Neither
$x^2$	$x^3$	$x^3 + x^2$
$x^4$	$x^5$	$e^x$
$\vert x\vert $	$x$	$\ln x$
$\cos x$	$\sin x$	$x + 1$
$1$ (constant)	$\frac{1}{x}$	$2^x$

Pattern: Even powers give even functions. Odd powers give odd functions. Mixing even and odd powers usually gives neither.

Increasing and Decreasing Functions

Definitions:

A function $f$ is:

Increasing on an interval if: whenever $x_1 < x_2$, then $f(x_1) < f(x_2)$
Decreasing on an interval if: whenever $x_1 < x_2$, then $f(x_1) > f(x_2)$

Graphically:

Increasing: the graph rises as you move right
Decreasing: the graph falls as you move right

    Increasing                      Decreasing

        y                               y
        |        ●                  ●   |
        |      /                      \ |
        |    /                          \
        |  ●                              \●
        +------→ x                    +------→ x

    "Going uphill"                  "Going downhill"

Practice Problems

Level 1 Recognizing Symmetry from a Graph

For each description, identify whether the function is even, odd, or neither:

A parabola opening upward with vertex at the origin
A straight line passing through the origin with positive slope
A straight line with $y$-intercept at $(0, 3)$

Thought Process

For each graph, imagine:

Folding along the $y$-axis (even test)
Rotating 180° around the origin (odd test)

Show Answer

(a) Even. A parabola with vertex at origin (like $y = x^2$) is symmetric about the $y$-axis.

(b) Odd. A line through the origin (like $y = mx$) is symmetric about the origin: rotate 180° and it looks identical.

(c) Neither. A line with non-zero $y$-intercept (like $y = x + 3$) has no symmetry about the $y$-axis or origin.

Level 2 Algebraic Symmetry Test

Determine whether each function is even, odd, or neither:

$f(x) = x^4 - 3x^2 + 1$
$g(x) = x^3 - x$
$h(x) = x^3 + 1$

Thought Process

For each function:

Compute $f(-x)$ by substituting $-x$ for every $x$
Simplify using the fact that $(-x)^n = x^n$ if $n$ is even, and $(-x)^n = -x^n$ if $n$ is odd
Compare to $f(x)$ and $-f(x)$

Show Answer

(a) $f(x) = x^4 - 3x^2 + 1$

$f(-x) = (-x)^4 - 3(-x)^2 + 1 = x^4 - 3x^2 + 1 = f(x)$

Even (all powers are even)

(b) $g(x) = x^3 - x$

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

Odd (all powers are odd)

(c) $h(x) = x^3 + 1$

$h(-x) = (-x)^3 + 1 = -x^3 + 1$

Is this equal to $h(x) = x^3 + 1$? No. Is this equal to $-h(x) = -x^3 - 1$? No.

Neither

Level 3 Intervals of Increase/Decrease

From the graph of $f(x) = x^3 - 3x$, determine:

The intervals where $f$ is increasing
The intervals where $f$ is decreasing
Is $f$ even, odd, or neither?

(Hint: The function has local maximum at $x = -1$ and local minimum at $x = 1$.)

Thought Process

Use the given critical points to divide the real line into intervals. On each interval, determine if the graph is rising or falling.

For symmetry: test algebraically by computing $f(-x)$.

Show Answer

(a) Increasing: $(-\infty, -1) \cup (1, \infty)$

The function rises from $-\infty$ until $x = -1$, then rises again after $x = 1$.

(b) Decreasing: $(-1, 1)$

Between the local max at $x = -1$ and local min at $x = 1$, the graph falls.

(c) Symmetry test:

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

Odd. (Notice the graph is symmetric about the origin: the local max at $(-1, 2)$ mirrors the local min at $(1, -2)$.)

Level 4 Products and Quotients of Even/Odd Functions

Let $f$ be an even function and $g$ be an odd function. Determine whether each of the following is even, odd, or cannot be determined:

$f \cdot g$ (the product)
$f / g$ (the quotient, where $g(x) \neq 0$)
$f \circ g$ (the composition $f(g(x))$)
$g \circ f$ (the composition $g(f(x))$)

Thought Process

For each operation, compute how it behaves when you input $-x$:

$f(-x) = f(x)$ (even)
$g(-x) = -g(x)$ (odd)

Then check if the result equals the original or its negative.

Show Answer

(a) Product $f \cdot g$: $(f \cdot g)(-x) = f(-x) \cdot g(-x) = f(x) \cdot (-g(x)) = -f(x)g(x) = -(f \cdot g)(x)$

Odd

(b) Quotient $f / g$: $(f/g)(-x) = \frac{f(-x)}{g(-x)} = \frac{f(x)}{-g(x)} = -\frac{f(x)}{g(x)} = -(f/g)(x)$

Odd

(c) Composition $f \circ g$: $(f \circ g)(-x) = f(g(-x)) = f(-g(x)) = f(g(x))$ (since $f$ is even: $f(-y) = f(y)$)

$= (f \circ g)(x)$

Even

(d) Composition $g \circ f$: $(g \circ f)(-x) = g(f(-x)) = g(f(x))$ (since $f$ is even)

$= (g \circ f)(x)$

Even

Level 5 Decomposition into Even and Odd Parts

Any function $f$ defined on a symmetric interval can be written as the sum of an even function and an odd function.

Show that for any function $f$, the expression $E(x) = \frac{f(x) + f(-x)}{2}$ is even.
Show that for any function $f$, the expression $O(x) = \frac{f(x) - f(-x)}{2}$ is odd.
Verify that $f(x) = E(x) + O(x)$.
Find the even and odd parts of $f(x) = e^x$.

Thought Process

To show $E(x)$ is even, compute $E(-x)$ and show it equals $E(x)$.

To show $O(x)$ is odd, compute $O(-x)$ and show it equals $-O(x)$.

For part (d), apply the formulas to $f(x) = e^x$.

Show Answer

(a) Show $E(x)$ is even: $$E(-x) = \frac{f(-x) + f(-(-x))}{2} = \frac{f(-x) + f(x)}{2} = E(x) \checkmark$$

(b) Show $O(x)$ is odd: $$O(-x) = \frac{f(-x) - f(-(-x))}{2} = \frac{f(-x) - f(x)}{2} = -\frac{f(x) - f(-x)}{2} = -O(x) \checkmark$$

(c) Verify $f(x) = E(x) + O(x)$: $$E(x) + O(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2} = \frac{2f(x)}{2} = f(x) \checkmark$$

(d) Decompose $f(x) = e^x$:

Even part: $$E(x) = \frac{e^x + e^{-x}}{2} = \cosh(x)$$

Odd part: $$O(x) = \frac{e^x - e^{-x}}{2} = \sinh(x)$$

So $e^x = \cosh(x) + \sinh(x)$, which decomposes the exponential into the hyperbolic cosine (even) and hyperbolic sine (odd).

Summary Table: Even vs. Odd

Property	Even	Odd
Test	$f(-x) = f(x)$	$f(-x) = -f(x)$
Graph symmetry	$y$-axis	Origin
Contains constant term?	Can	Cannot (why?)
Value at $x = 0$	Any	Must be $f(0) = 0$
Examples	$x^2$, $\cos x$, $\vert x\vert $	$x^3$, $\sin x$, $\frac{1}{x}$

Note: If $f$ is odd and $f(0)$ is defined, then $f(0) = 0$. (Proof: $f(0) = f(-0) = -f(0)$, so $2f(0) = 0$.)

Common Misconceptions

Common misconception

a function is even if its graph looks like a “smooth” curve. Even symmetry has a precise algebraic meaning: $f(-x) = f(x)$ for all $x$ in the domain. A function can have a smooth, pleasant-looking graph without any symmetry at all. Conversely, $f(x) = x^4 - 2x^2 + 1$ is even (satisfies the algebraic test) even though it has multiple humps. Symmetry must be verified algebraically, not inferred from visual impression alone.

Common misconception

every function is either even or odd. Most functions are neither. For example, $f(x) = x + 1$ satisfies $f(-x) = -x + 1$, which is neither $f(x)$ nor $-f(x)$ (unless $x = 0$). A function is even only if $f(-x) = f(x)$ everywhere, and odd only if $f(-x) = -f(x)$ everywhere. Failing both tests means the function has no simple reflection symmetry.

Mastery Checklist

I can test algebraically whether a function is even, odd, or neither
I can identify symmetry from a graph
I can identify intervals where a function is increasing or decreasing
I understand that mixing even and odd powers usually gives neither
I can work with products and compositions of even/odd functions

Mental Model

The Mirror and Rotation Tests:

Even (Mirror): Stand a mirror on the $y$-axis. If the reflection looks exactly like the original, the function is even.
Odd (Rotation): Put a pin at the origin and spin the graph 180°. If it looks exactly the same, the function is odd.

If neither test works, the function has no special symmetry.

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Last updated: 2026-01-22