Odd and even functions

Q: What defines the graphical symmetry of an even function?

An even function is perfectly symmetrical about the y-axis , meaning the left side is a mirror image of the right side.

Q: Is the absolute value function \(f(x) = |x|\) even, odd, or neither?

The absolute value function is an even function because \(f(-x) = |-x| = |x| = f(x)\).

Q: How can you identify an even polynomial function by looking at its powers?

A polynomial is even if all the powers of \(x\) are even numbers (e.g., 2, 4, 6) and any constant term is considered an even power (\(x^0\)).

Q: What defines the graphical symmetry of an odd function?

An odd function is symmetrical about the origin (0,0) , which is a 180° rotational symmetry.

Q: How can you identify an odd polynomial function by looking at its powers?

A polynomial is odd if all powers of \(x\) are odd numbers (e.g., 1, 3, 5) and there is no constant term .

Q: Which trigonometric functions are even and which are odd?

Cosine is an even function (\(\cos(-x) = \cos(x)\)), and Sine is an odd function (\(\sin(-x) = -\sin(x)\)).

Practice

Mathematics M2 (Algebra and Calculus) · Senior Secondary (HKDSE) · 6 min read · Updated: 17 Aug 2025

Odd and Even Functions: Your Guide to Function Symmetry!

Hello! Welcome to our study notes on Odd and Even Functions. Don't worry if this sounds a bit strange – we're not talking about numbers, but a special property of functions. Think of it as giving functions a personality: some are perfectly balanced and symmetrical (even), while others have a cool, rotational balance (odd).

Understanding this concept is super useful in M2. It helps you predict what a graph will look like and can be a massive shortcut in calculus problems later on. So, let's get started!

What is an Even Function? The "Butterfly" Symmetry

Imagine a butterfly. If you fold its wings along its body, the left and right sides match perfectly. Even functions are like that! Their graphs are perfectly symmetrical about the y-axis.

This means that the part of the graph on the left side of the y-axis is a mirror image of the part on the right side.

The Graphical Test

If you can "fold" the graph along the y-axis and the two halves match up perfectly, it's an even function. Look at the graph of \(f(x) = x^2\) below. It's a classic example of an even function.

The Algebraic Test (The Formal Definition)

While looking at a graph is helpful, we need a solid mathematical way to prove it. Here’s the rule:

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

What does this mean? It means if you plug in a negative value for x, you get the exact same output as if you plugged in the positive version of that value. Let's test this with an example.

Example: Is \(f(x) = x^4 - 2x^2\) an even function?

Step 1: Write down the original function, \(f(x)\).
\(f(x) = x^4 - 2x^2\)

Step 2: Find \(f(-x)\) by replacing every single 'x' with '(-x)'. Always use brackets! This is super important.
\(f(-x) = (-x)^4 - 2(-x)^2\)

Step 3: Simplify the expression for \(f(-x)\). Remember that a negative number raised to an even power becomes positive.
\(f(-x) = (x^4) - 2(x^2)\) \(f(-x) = x^4 - 2x^2\)

Step 4: Compare your result with the original function \(f(x)\).
We found that \(f(-x) = x^4 - 2x^2\).
The original function was \(f(x) = x^4 - 2x^2\).
They are exactly the same! Since \(f(-x) = f(x)\), the function is even.

A Special Example: The Absolute Value Function

As required by the syllabus, you need to know that the absolute value function, \(f(x) = |x|\), is an even function.

Graphically: The graph of \(f(x) = |x|\) is a V-shape with its point at the origin. It's perfectly symmetric about the y-axis.
Algebraically: Let's test it.
\(f(x) = |x|\)
\(f(-x) = |-x|\)
Since the absolute value of a negative number is its positive counterpart, \(|-x| = |x|\).
So, \(f(-x) = f(x)\). It's confirmed to be an even function!

Memory Aid for Even Functions

Think of polynomial functions. If all the powers of x are even numbers (like in \(f(x) = 3x^6 - x^2 + 5\)), the function is often even. The constant term (like +5) can be thought of as \(5x^0\), and 0 is an even number!

Key Takeaway for Even Functions

Algebraic Test: \(f(-x) = f(x)\)
Symmetry: Symmetric about the y-axis (mirror image).
Classic Examples: \(f(x) = x^2\), \(f(x) = \cos(x)\), \(f(x) = |x|\)

What is an Odd Function? The "Pinwheel" Symmetry

Odd functions have a different kind of balance. They are symmetrical about the origin (0,0).

Think of it like a pinwheel. If you rotate the graph 180° around the origin, it will look exactly the same as it did before you rotated it.

The Graphical Test

If you can rotate the graph 180° about the origin and it lands back on itself, it's an odd function. Check out the graph of \(f(x) = x^3\). It has this cool rotational symmetry.

The Algebraic Test (The Formal Definition)

Here’s the rule for odd functions:

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

This means that if you plug in a negative value for x, you get the negative of the output you would get from plugging in the positive version.

Example: Is \(f(x) = x^3 - 4x\) an odd function?

Step 1: Write down the original function, \(f(x)\).
\(f(x) = x^3 - 4x\)

Step 2: Find \(f(-x)\) by replacing every 'x' with '(-x)'.
\(f(-x) = (-x)^3 - 4(-x)\)

Step 3: Simplify the expression for \(f(-x)\). Remember that a negative number raised to an odd power stays negative.
\(f(-x) = -x^3 + 4x\)

Step 4: Compare \(f(-x)\) with \(-f(x)\). To find \(-f(x)\), just multiply the entire original function by -1.
\(-f(x) = -(x^3 - 4x) = -x^3 + 4x\)
Look! Our result for \(f(-x)\) is exactly the same as \(-f(x)\)!
Since \(f(-x) = -f(x)\), the function is odd.

Memory Aid for Odd Functions

You guessed it! For polynomial functions, if all the powers of x are odd numbers (like in \(f(x) = 2x^5 - 9x^3 + x\)) and there is no constant term, the function is often odd.

Key Takeaway for Odd Functions

Algebraic Test: \(f(-x) = -f(x)\)
Symmetry: Symmetric about the origin (180° rotational symmetry).
Classic Examples: \(f(x) = x^3\), \(f(x) = x\), \(f(x) = \sin(x)\)

What if a Function is Neither?

This is a really important point: most functions are neither odd nor even!

Don't fall into the trap of thinking a function must be one or the other. If a function fails both the even test and the odd test, it's simply "neither".

Example: Is \(f(x) = x^2 + 3x\) odd, even, or neither?

Step 1: Find \(f(-x)\).
\(f(-x) = (-x)^2 + 3(-x)\) \(f(-x) = x^2 - 3x\)

Step 2: Check if it's even. Is \(f(-x) = f(x)\)?
Is \(x^2 - 3x\) the same as \(x^2 + 3x\)? No. So, it's not even.

Step 3: Check if it's odd. Is \(f(-x) = -f(x)\)?
First, find \(-f(x) = -(x^2 + 3x) = -x^2 - 3x\).
Is \(x^2 - 3x\) the same as \(-x^2 - 3x\)? No. So, it's not odd.

Conclusion: Since the function is not even and not odd, it is neither. Its graph will not have y-axis symmetry or origin symmetry.

Key Takeaway for "Neither"

Algebraic Test: \(f(-x)\) is not equal to \(f(x)\) AND it is not equal to \(-f(x)\).
Symmetry: No y-axis or origin symmetry.
Important: This is the most common category for functions!

Quick Summary and The Big Picture

Here is everything in a simple table. Use this as a quick review!

	Even Function	Odd Function	Neither
Algebraic Test	\(f(-x) = f(x)\)	\(f(-x) = -f(x)\)	Fails both tests
Graph Symmetry	Symmetric about the y-axis	Symmetric about the origin	No special symmetry
Example	\(f(x) = x^2\), \(f(x) =	x	\)	\(f(x) = x^3\), \(f(x) = x\)	\(f(x) = x+1\), \(f(x) = \sqrt{x}\)

Did you know?

The trigonometric functions you use all the time have these properties!
- Cosine is an even function: \(\cos(-x) = \cos(x)\)
- Sine is an odd function: \(\sin(-x) = -\sin(x)\)
This knowledge will become very powerful when you study definite integrals in Calculus. For example, integrating an odd function over a symmetric interval (like from -5 to 5) always gives an answer of zero. It's an amazing shortcut!

Great job getting through this topic! Take your time, practice the algebraic tests, and you'll be a pro at identifying odd and even functions in no time.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What defines the graphical symmetry of an even function?

An even function is perfectly symmetrical about the y-axis, meaning the left side is a mirror image of the right side.

What is the formal algebraic definition of an even function?

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

Is the absolute value function \(f(x) = |x|\) even, odd, or neither?

The absolute value function is an even function because \(f(-x) = |-x| = |x| = f(x)\).

How can you identify an even polynomial function by looking at its powers?

A polynomial is even if all the powers of \(x\) are even numbers (e.g., 2, 4, 6) and any constant term is considered an even power (\(x^0\)).

What defines the graphical symmetry of an odd function?

An odd function is symmetrical about the origin (0,0), which is a 180° rotational symmetry.

What is the formal algebraic definition of an odd function?

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

How can you identify an odd polynomial function by looking at its powers?

A polynomial is odd if all powers of \(x\) are odd numbers (e.g., 1, 3, 5) and there is no constant term.

What is required for a function to be classified as "neither" odd nor even?

A function is "neither" if it fails both tests: \(f(-x)\) is not equal to \(f(x)\) and \(f(-x)\) is not equal to \(-f(x)\).

Which trigonometric functions are even and which are odd?

Cosine is an even function (\(\cos(-x) = \cos(x)\)), and Sine is an odd function (\(\sin(-x) = -\sin(x)\)).

What is the calculus shortcut for integrating an odd function over a symmetric interval?

The definite integral of an odd function from \(-a\) to \(a\) is always zero.

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