Introduction: Symmetry in Functions

Some functions possess beautiful symmetry properties that simplify analysis, graphing, and problem-solving. In this subtopic, we explore odd and even functions (defined by their geometric symmetry), inverse functions and when they exist, domain restriction as a tool to create invertible functions, and the special class of self-inverse functions.

These ideas connect algebra, geometry, and the fundamental concept of what a function actually does — making them a rich and examinable area of the AHL course.

Even Functions

Even Function

A function $f (x)$ is even if $f (- x) = f (x)$ for all $x$ in the domain of $f$ . Geometrically, even functions are symmetric about the y-axis.

To verify a function is even, substitute $- x$ for $x$ and show the expression simplifies back to $f (x)$ .

Even Functions

Example

Show that $f (x) = x^{4} - 3 x^{2} + 5$ is an even function.

$f (- x) = (- x)^{4} - 3 (- x)^{2} + 5 = x^{4} - 3 x^{2} + 5 = f (x) ✓$

Since $f (- x) = f (x)$ , the function is even. Its graph is symmetric about the y-axis.

Exam Tip

A polynomial function is even if and only if all its terms have even powers (including constant terms, which are degree 0 — an even number). This gives a quick visual check.

Common even functions to recognise:

$f (x) = x^{2}$ , $f (x) = x^{4}$ , $f (x) = ∣ x ∣$
$f (x) = cos x$ (the cosine function is even)
Any constant function $f (x) = c$

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AHL 2.14—Odd and even functions, self-inverse, inverse and domain restriction

Introduction: Symmetry in Functions

Even Functions

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