Function Symmetry

MATH161

Reference: Stewart 1.1 • Chapter: 1 • Section: 1

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Function Symmetry

Finding Shortcuts Through 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 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 $f(x)$:

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 $2f(0) = 0$.)

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