Topic 1.5B · CED: Polynomial Functions and Complex Zeros

Even and Odd Functions

10 MIN READ12 IDEAS22 PROBLEMS14 flashcards

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01
To test: substitute $-x$ into the function. If you get the same as the original function, it's even.
02
To test: substitute $-x$ into the function. If you get the opposite (negative) of the original, it's odd.
03
Even: $f(-x) = f(x)$ — the value at $-x$ is the same.

Odd: $f(-x) = -f(x)$ — the value at $-x$ is the opposite.

To test: substitute $-x$ into the function. If you get the same as the original function, it's even.
To test: substitute $-x$ into the function. If you get the opposite (negative) of the original, it's odd.
Even: $f(-x) = f(x)$ — the value at $-x$ is the same.

Odd: $f(-x) = -f(x)$ — the value at $-x$ is the opposite.
If $f$ is odd and $f(0)$ is defined, then $f(0) = 0$ .

Why: putting $x = 0$ into $f(-x) = -f(x)$ gives $f(0) = -f(0)$ , and the only number equal to its own negative is $0$ .
Sums (only when both pieces are the same kind):

even $+$ even $=$ even

odd $+$ odd $=$ odd

even $+$ odd $=$ Usually neither
Products — treat even as $+$ and odd as $-$ :

even $\times$ even $=$ even

odd $\times$ odd $=$ Even

even $\times$ odd $=$ odd

Practice 22 problems Next: Chapter 1.6 Polynomial Functions and End Behavior

Even Functions Are Symmetrical Over the y-Axis

EVEN functions are symmetrical over the y-axis (the line $x = 0$ ). This works for all kinds of shapes — not just parabolas!

Three completely different shapes — all even, because all are mirror-symmetric about the y-axis.

CONCEPT

An EVEN Function...

is graphically symmetric over the line $x = 0$ .

analytically has the property: $f(-x) = f(x)$

KEY RULE

To test: substitute $-x$ into the function. If you get the SAME AS the original function, it's EVEN.

Odd Functions Are Symmetrical Over the Origin

ODD functions are symmetrical over the origin, the point $(0, 0)$ . Picture spinning the graph 180° around the center — an odd function looks exactly the same after the spin.

Three different shapes — all odd, because all look unchanged after a 180° spin about the origin.

CONCEPT

An ODD Function...

is graphically symmetric over the point $(0, 0)$ .

analytically has the property: $f(-x) = -f(x)$

KEY RULE

To test: substitute $-x$ into the function. If you get the OPPOSITE (negative) of the original, it's ODD.

A Fast Shortcut for Polynomials: Just Look at the Exponents

Here's a huge time-saver that works specifically for polynomials — before you do any substitution at all, just glance at the exponents:

CONCEPT

The Exponent Shortcut

EVERY exponent is EVEN (including a constant term, which is $x^{0}$ ) → the function is EVEN.

EVERY exponent is ODD → the function is ODD.

The exponents are MIXED (some even, some odd) → the function is usually NEITHER.

COMMON MISTAKE

The shortcut is exact — but only once the polynomial is EXPANDED. There are no counterexamples for an expanded polynomial.

The real trap is judging a FACTORED form without expanding it. $x(x^2 - 9)$ looks mixed, but expand it to $x^3 - 9x$ and every exponent is odd — the function is odd.

Remember: a plain number (a constant term, like the $+3$ in $x^2+3$ ) counts as an EVEN exponent, since it is really $x^0$ .

Quick check

Is $h(x) = x^{5} - 3x^{3} + 2x$ even, odd, or neither?

Worked Example — Proving a Function Is Even

Worked example

Show analytically that $f(x) = x^{6} - 4x^{2}$ is an even function.

01
Quick check first: the exponents are 6 and 2 — both EVEN. That's a strong hint this function is even.
02
Now confirm it properly. Substitute $-x$ in for every $x$ :
$f(-x) = (-x)^6 - 4(-x)^2 = x^6 - 4x^2$
03
Compare the result to the original $f(x)$ :
$f(-x) = x^6 - 4x^2 = f(x)$
04
Since $f(-x) = f(x)$ , the function IS even — confirmed both by the shortcut and by the algebra.

Worked Example — Proving a Function Is Odd

Worked example

Show analytically that $f(x) = -2x^{3} + 5x$ is an odd function.

01
Quick check first: the exponents are 3 and 1 — both ODD. This should turn out to be an odd function.
02
Substitute $-x$ in for every $x$ :
$f(-x) = -2(-x)^3 + 5(-x) = 2x^3 - 5x$
03
Now compute $-f(x)$ (negate the ENTIRE original function) so we have something to compare against:
$-f(x) = -(-2x^3 + 5x) = 2x^3 - 5x$
04
Compare $f(-x)$ to $-f(x)$ :
$f(-x) = 2x^3 - 5x = -f(x)$
05
Since $f(-x) = -f(x)$ , the function IS odd.

Worked Example — Neither Even nor Odd

Worked example

Show analytically whether $f(x) = 6x^{4} - 2x$ is even, odd, or neither.

01
Quick check first: the exponents are 4 (even) and 1 (odd) — MIXED. This is a strong signal it will be neither.
02
Substitute $-x$ in for every $x$ :
$f(-x) = 6(-x)^4 - 2(-x) = 6x^4 + 2x$
03
Write out $f(x)$ and $-f(x)$ so you have both to compare against:
$f(x) = 6x^4 - 2x \qquad -f(x) = -6x^4 + 2x$
04
Compare: $f(-x) = 6x^{4}+2x$ . Is this the same as $f(x) = 6x^{4}-2x$ ? No. Is it the same as $-f(x) = -6x^{4}+2x$ ? Also no.
05
Since $f(-x)$ matches NEITHER $f(x)$ NOR $-f(x)$ , this function is NEITHER even nor odd.

Testing from a Graph

CONCEPT

Visual Test

Fold the graph along the y-axis — if both halves match exactly, it's even.

Rotate the graph 180° around the origin — if it looks unchanged, it's odd.

If neither test works, the function is neither even nor odd.

Testing from a Table of Values

Worked example

Use the table to decide: even, odd, or neither?

x	f(x)
−2	8
−1	1
0	0
1	−1
2	−8

01
Compare $f(-1)$ to $f(1)$ : $f(-1)=1$ , $f(1)=-1$ , so $f(-1) = -f(1)$ .
02
Compare $f(-2)$ to $f(2)$ : $f(-2)=8$ , $f(2)=-8$ , so $f(-2) = -f(2)$ .
03
Since $f(-x) = -f(x)$ for every pair, this function is ODD.

Using Symmetry to Fill In Missing Values

Once you know a function is even or odd, symmetry hands you values you were never given. Tables and graphs on the exam are often half-blank on purpose.

KEY RULE

Even: $f(-x) = f(x)$ — the value at $-x$ is the SAME.

Odd: $f(-x) = -f(x)$ — the value at $-x$ is the OPPOSITE.

KEY RULE

If $f$ is ODD and $f(0)$ is defined, then $f(0) = 0$ .

Why: putting $x = 0$ into $f(-x) = -f(x)$ gives $f(0) = -f(0)$ , and the only number equal to its own negative is $0$ .

Worked example

$g$ is even. The table gives $g(1) = 4$ , $g(3) = -2$ , $g(5) = 7$ . Find $g(-3)$ and $g(-5)$ .

01
Even means the sign of the input does not matter:
$g(-x) = g(x)$
02
Copy the values straight across:
$g(-3) = g(3) = -2 \qquad g(-5) = g(5) = 7$

Worked example

$h$ is odd and continuous. The table gives $h(2) = -6$ and $h(4) = 1$ . Find $h(-2)$ , $h(-4)$ , and $h(0)$ .

01
Odd means flip the sign of the output:
$h(-x) = -h(x)$
02
Flip each known value:
$h(-2) = -h(2) = -(-6) = 6 \qquad h(-4) = -h(4) = -1$
03
$h$ is odd and defined at $0$ , so $h(0) = 0$ .

COMMON MISTAKE

Flipping the sign for an EVEN function. Even functions copy the value; only odd functions flip it.

Writing $f(0) = 0$ for an even function. That is only forced for odd ones — an even function can have any value at $0$ .

Reading It Off a Graph

The same rule works with a picture. For an even function, whatever the graph does at $x = 3$ it does at $x = -3$ at the same height. For an odd function the point at $x = -3$ sits at the same distance from the axis but on the other side.

CONCEPT

Half a Graph Is a Whole Graph

Given the right half and the word "even", reflect it across the $y$ -axis.

Given the right half and the word "odd", rotate it $180°$ about the origin.

Adding and Multiplying Even and Odd Functions

Build a new function out of two you already know, and its symmetry follows a fixed pattern. Multiplication behaves exactly like the sign rules you already know for positive and negative numbers.

KEY RULE

Sums (only when BOTH pieces are the same kind):

even $+$ even $=$ even

odd $+$ odd $=$ odd

even $+$ odd $=$ usually NEITHER

KEY RULE

Products — treat even as $+$ and odd as $-$ :

even $\times$ even $=$ even

odd $\times$ odd $=$ EVEN

even $\times$ odd $=$ odd

CONCEPT

Where the Product Rule Comes From

Take $f$ odd and $g$ odd. Then

f(-x)\,g(-x) = \bigl(-f(x)\bigr)\bigl(-g(x)\bigr) = f(x)\,g(x)

Two minus signs cancel, so the product is even. That is why $x \cdot x = x^2$ — odd times odd — comes out even.

Worked example

$f$ is odd and $g$ is even. Classify $f(x)g(x)$ and $f(x) + g(x)$ .

01
For the product, substitute $-x$ :
$f(-x)\,g(-x) = \bigl(-f(x)\bigr)\bigl(g(x)\bigr) = -f(x)\,g(x)$
02
That is the opposite of $f(x)g(x)$ , so the product is ODD.
03
For the sum:
$f(-x) + g(-x) = -f(x) + g(x)$
04
That is neither $f(x) + g(x)$ nor its negative, so the sum is NEITHER — unless one of the two functions is just $0$ .

COMMON MISTAKE

Assuming odd $\times$ odd stays odd. It turns EVEN — the two sign flips cancel. This is the single most-missed line in this topic.

Expecting even $+$ odd to be something. Mixing the two kinds is exactly how a function ends up neither — which is why $x^2 + x$ is neither.

Quick check

$f$ is odd and $g$ is odd. Is $f \cdot g$ even, odd, or neither?

Practice Problems

Worked example

Problem 1. Is $f(x) = x^{4} - 2x^{2}$ even, odd, or neither? (Try the exponent shortcut first!)

01
Exponents are 4 and 2 — both even → predict EVEN.
02
Confirm: $f(-x) = (-x)^{4} - 2(-x)^{2} = x^{4} - 2x^{2} = f(x)$ . Confirmed EVEN.

Worked example

Problem 2. Is $g(x) = x^{3} + 2$ even, odd, or neither?

01
Exponents are 3 (odd) and 0 (even, since the constant 2 = $2x^{0}$ ) — MIXED → predict NEITHER.
02
Confirm: $g(-x) = (-x)^{3}+2 = -x^{3}+2$ . Compare to $g(x)=x^{3}+2$ (not equal) and $-g(x)=-x^{3}-2$ (not equal either). Confirmed NEITHER.

Common slips

The shortcut is exact — but only once the polynomial is expanded. There are no counterexamples for an expanded polynomial.

The real trap is judging a factored form without expanding it. $x(x^2 - 9)$ looks mixed, but expand it to $x^3 - 9x$ and every exponent is odd — the function is odd.

Remember: a plain number (a constant term, like the $+3$ in $x^2+3$ ) counts as an even exponent, since it is really $x^0$ .
Flipping the sign for an even function. Even functions copy the value; only odd functions flip it.

Writing $f(0) = 0$ for an even function. That is only forced for odd ones — an even function can have any value at $0$ .
Assuming odd $\times$ odd stays odd. It turns even — the two sign flips cancel. This is the single most-missed line in this topic.

Expecting even $+$ odd to be something. Mixing the two kinds is exactly how a function ends up neither — which is why $x^2 + x$ is neither.

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14 cards · Polynomials, Zeros and multiplicity

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Recap card

5 lines to re-read the night before.

01
Even: $f(-x) = f(x)$ , symmetric about the y-axis (line $x=0$ ) — parabolas, V-shapes, and cosine waves are all classic examples.
02
Odd: $f(-x) = -f(x)$ , symmetric about the origin $(0, 0)$ — lines through the origin, S-curve cubics, and sine waves are all classic examples.
03
Shortcut for polynomials: all exponents even → even; all exponents odd → odd; mixed → usually neither.
04
Always confirm algebraically by substituting $-x$ and comparing to both $f(x)$ and $-f(x)$ .
05
Many functions — especially ones with mixed-parity exponents — are neither even nor odd.

Practice 22 problems Next: Chapter 1.6 Polynomial Functions and End Behavior

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Even and Odd Functions

Even Functions Are Symmetrical Over the y-Axis

Odd Functions Are Symmetrical Over the Origin

A Fast Shortcut for Polynomials: Just Look at the Exponents

Worked Example — Proving a Function Is Even

Worked Example — Proving a Function Is Odd

Worked Example — Neither Even nor Odd

Testing from a Graph

Testing from a Table of Values

Using Symmetry to Fill In Missing Values

Reading It Off a Graph

Adding and Multiplying Even and Odd Functions

Practice Problems

Recap card

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