Algebra • Topic 4

Functions

In Olympiad mathematics, functions are often treated more abstractly than in standard calculus. We focus on properties like injectivity, surjectivity, and solving functional equations.

Definitions

Definition (Function)

A function

f : A \to B

is a rule that assigns exactly one element

f (x) \in B

to every element

x \in A

$A$ is the domain.
$B$ is the codomain.
The set of actual outputs ${f (x) ∣ x \in A}$ is the range (or image).

Function Properties

1. Parity

Even Function: $f (- x) = f (x)$ for all $x$ . (Symmetric about y-axis).
Odd Function: $f (- x) = - f (x)$ for all $x$ . (Symmetric about origin).

2. Monotonicity

Increasing: $x < y ⟹ f (x) \leq f (y)$ .
Strictly Increasing: $x < y ⟹ f (x) < f (y)$ .

Functional Equations (Intro)

A functional equation asks you to find all functions

f

that satisfy a given relation.

Example

Find all functions

f : R \to R

such that

f (x + y) = f (x) + f (y)

for all

x, y

. (Cauchy's Equation)

Note

Without continuity or other constraints, this has "wild" solutions. However, if we assume

f

is continuous, monotonic, or bounded, the only solutions are linear:

f (x) = c x

Basic Substitution Tactics

To solve simple functional equations, substitute specific values for variables.

Let $x = 0$ or $x = 1$ .
Let $y = x$ or $y = - x$ .
If the domain allows, let $x \to 1/ x$ .

Example (Finding Values)

f (x) + 2 f (\frac{1}{x}) = 3 x

for all

x \neq = 0

, find

f (x)

Proof (Solution). 1. Original equation: $f (x) + 2 f (\frac{1}{x}) = 3 x$

Replace $x$ with $1/ x$ : $f (\frac{1}{x}) + 2 f (x) = \frac{3}{x}$
We now have a system of two linear equations for variables $A = f (x)$ and $B = f (1/ x)$ :

$A + 2 B = 3 x$

$2 A + B = \frac{3}{x}$

4. Multiply the second eq by 2:

4 A + 2 B = \frac{6}{x}

Subtract the first eq: $(4 A + 2 B) - (A + 2 B) = \frac{6}{x} - 3 x ⟹ 3 A = \frac{6}{x} - 3 x$ .
$A = \frac{2}{x} - x$ .

Therefore,

f (x) = \frac{2}{x} - x

. ∎

Practice Problems

Exercise (Problem 1)

Let

f (x) = \frac{x}{1 - x}

. Find a formula for the composite function

(f \circ f) (x)

Exercise (Problem 2)

Determine if

f (x) = x^{3} + x

is odd, even, or neither.

Exercise (Problem 3)

Find all functions

f : R \to R

such that

2 f (x) - f (1 - x) = x^{2}