Functions, Relations & Functional Equations

Axiomatic Definition

A function $f: A \to B$ is a binary relation that associates each element $x \in A$ with exactly one element $f(x) \in B$. $A$ is the domain, $B$ is the codomain, and $f(A) = \{f(x) : x \in A\} \subseteq B$ is the range.

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1. Classification of Mappings

• Injective (One-to-One): $f(x_1)=f(x_2)⟹x_1=x_2$. (Geometrically: Any horizontal line intersects the graph at most once; derivative $f^{\prime }(x)>0$ or $f^{\prime }(x)<0$ strictly monotonic).
• Surjective (Onto): $\text{Range}(f)=\text{Codomain}(B)$.
• Bijective (Invertible): Both injective and surjective $⟺$ Inverse function $f^{-1}:B\to A$ exists uniquely.

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2. Parity, Periodicity & Transformations

Even and Odd Functions

• Even: $f(-x)=f(x)$ $⟹$ Symmetric about the $y$-axis; ${\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx$.
• Odd: $f(-x)=-f(x)$ $⟹$ Symmetric about the origin (rotation by $\pi$); ${\int }_{-a}^{a}f(x)dx=0$.

Periodicity

A function $f(x)$ has fundamental period $T>0$ if $f(x+T)=f(x)$ for all $x\in \text{Dom}(f)$.

• If $f(x)$ has period $T$, then $f(ax+b)$ has period $\frac{T}{|a|}$.
• If $f(x+a)=-f(x)$, then $f(x+2a)=-f(x+a)=f(x)⟹T=2a$.
• If $f(x+a)=\frac{1}{f(x)}$, then $f(x+2a)=f(x)⟹T=2a$.

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3. Classical Functional Equations

Cauchy's Classical Functional Equations (for continuous $f$)

$$\begin{aligned} f(x + y) &= f(x) + f(y) \implies f(x) = k x \\ f(x + y) &= f(x) f(y) \implies f(x) = a^x = e^{k x} \\ f(x y) &= f(x) + f(y) \implies f(x) = k \ln x \\ f(x y) &= f(x) f(y) \implies f(x) = x^n \end{aligned}$$

Polynomial Identity

If $P(x)$ is a polynomial satisfying $P(x)\cdot P\left(\frac{1}{x}\right)=P(x)+P\left(\frac{1}{x}\right)$, then:

$$P(x)=1\pm x^n(n\in \mathbb{N})$$