Function (mathematics) Study Guide

📖 Core Concepts Function: A rule assigning exactly one output \(y\) in the codomain \(Y\) to each input \(x\) in the domain \(X\); written \(f\colon X\to Y,\;f(x)=y\). Domain / Codomain: Set of allowed inputs vs. set that contains all possible outputs. Image (Range): Set of actual outputs \(\{f(x)\mid x\in X\}\). Preimage: For \(y\in Y\), the set \(\{x\in X\mid f(x)=y\}\); for a subset \(B\subseteq Y\), \(\{x\in X\mid f(x)\in B\}\). Partial vs. Total Function: Partial – each \(x\) relates to at most one \(y\); total – defined for every \(x\in X\). Injective (One‑to‑One): Different inputs give different outputs; \(f(x1)=f(x2)\Rightarrow x1=x2\). Surjective (Onto): Every \(y\in Y\) is hit by some \(x\in X\). Bijective: Both injective and surjective; admits a two‑sided inverse \(f^{-1}\). Composition: \((g\circ f)(x)=g(f(x))\); associative, identity function \(\operatorname{id}\) is neutral. Restriction / Extension: \(f|S\) limits the domain to \(S\subseteq X\); extensions (e.g., analytic continuation) enlarge the domain while preserving structure. Multivariate Function: Takes an ordered \(n\)-tuple \((x1,\dots,xn)\) as input: \(f\colon X1\times\cdots\times Xn\to Y\). Notation: Arrow \(x\mapsto x+1\); index \(an\) for sequences; power‑series, recurrence, implicit definitions. --- 📌 Must Remember Definition (Set‑theoretic): \(f\subseteq X\times Y\) with \(\forall x\in X\,\exists!\,y\in Y\;(x,y)\in f\). Injective ⇔ Left Inverse: \(\exists g\) s.t. \(g\circ f = \operatorname{id}X\). Surjective ⇔ Right Inverse (requires Choice): \(\exists g\) s.t. \(f\circ g = \operatorname{id}Y\). Bijective ⇔ Two‑Sided Inverse: \(f^{-1}\circ f = \operatorname{id}X\) and \(f\circ f^{-1} = \operatorname{id}Y\). Canonical Factorization: Any \(f\) = injection \(\circ\) surjection: \(f = i\circ s\) where \(s\colon X\to f(X)\) is onto and \(i\colon f(X)\hookrightarrow Y\) is one‑to‑one. Domain of a Formula: Remove points that make denominators zero or radicands negative. Monotonic ⇒ Invertible (on the image). Graph of \(f\): \(\{(x,f(x))\mid x\in X\}\). --- 🔄 Key Processes Checking Injectivity Assume \(f(x1)=f(x2)\). Manipulate algebraically to deduce \(x1=x2\). Checking Surjectivity Start with arbitrary \(y\in Y\). Solve \(f(x)=y\) for \(x\); show a solution exists for every \(y\). Finding the Inverse of a Bijective Function Write the equation \(y = f(x)\). Solve explicitly for \(x\) in terms of \(y\); rename the solution \(f^{-1}(y)\). Restricting a Function Choose subset \(S\subseteq X\). Define \(f|S(x)=f(x)\) for \(x\in S\). Composing Functions Verify codomain of the first matches domain of the second. Substitute: \((g\circ f)(x)=g(f(x))\). Canonical Factorization Compute image \(f(X)\). Define surjection \(s\colon X\to f(X)\), \(s(x)=f(x)\). Define injection \(i\colon f(X)\hookrightarrow Y\), \(i(y)=y\). --- 🔍 Key Comparisons Injective vs. Surjective Injective: No two distinct inputs share an output. Surjective: Every possible output is used at least once. Partial vs. Total Function Partial: May be undefined for some \(x\in X\). Total: Defined for every \(x\in X\). Restriction vs. Extension Restriction: Shrinks the domain, keeps rule unchanged. Extension: Enlarges the domain, often by a new rule that agrees on the overlap. Arrow Notation vs. Index Notation Arrow: Describes rule without naming the function (\(x\mapsto x+1\)). Index: Treats function as a sequence (\(an\)). --- ⚠️ Common Misunderstandings “Every function has an inverse.” Only bijections have a two‑sided inverse; injective‑only functions have a left inverse on the image, not on the whole codomain. Confusing image with codomain. The image may be a proper subset of the codomain; surjectivity means they coincide. Assuming composition is commutative. In general \(g\circ f \neq f\circ g\). Partial function = undefined everywhere else. It is still a relation; the domain of definition is just the set where a value exists. --- 🧠 Mental Models / Intuition Function as a “machine”: each input slides through a single‑exit pipe; think of the graph as the set of all (input,output) pins on a board. Injective = “no collisions”: imagine parking cars in distinct spots; two cars never share the same spot. Surjective = “full coverage”: every parking spot (output) has at least one car (input). Bijective = “perfect one‑to‑one matching” – a perfect shuffle of two decks. Restriction = “closing off doors”: you lock off parts of the domain; everything else works unchanged. --- 🚩 Exceptions & Edge Cases Empty Domain: The empty function \(f\colon\varnothing\to Y\) is vacuously injective and surjective onto the empty image, but not onto a non‑empty codomain. Multi‑valued “functions”: E.g., complex square root; require branch cuts to become true (single‑valued) functions. Axiom of Choice: Right‑inverse characterization of surjectivity relies on it; without Choice, a surjection may lack a global right inverse. Implicit Definitions: May define a function locally but not globally (e.g., \(x^2+y^2=1\) defines \(y=\pm\sqrt{1-x^2}\)). --- 📍 When to Use Which Prove bijectivity → Show both injective and surjective (or construct explicit inverse). Find inverse → Use algebraic solving when the function is given by a formula; use monotonicity for real functions. Determine domain of a formula → Look for division by zero, even roots of negative numbers, logarithms of non‑positive numbers. Choose notation → Arrow notation for ad‑hoc rule definitions; index notation for sequences; power‑series when convergence properties matter. Apply restriction → When you need a function only on a subset where a property (e.g., injectivity) holds, enabling a local inverse. --- 👀 Patterns to Recognize Linear‑fractional forms \(\frac{ax+b}{cx+d}\) often hint at a bijection on \(\mathbb{R}\setminus\{-d/c\}\). Quadratic equations in the definition of a function usually signal two‑valued behavior → need to pick a branch for a true function. Monotonicity + continuity on an interval ⇒ automatically surjective onto its image and invertible there. Composition associativity appears in nested function problems; you can regroup without changing the result. --- 🗂️ Exam Traps “If a function is injective, it must be surjective.” Wrong unless the codomain equals the image. Choosing the wrong branch of a multi‑valued function (e.g., picking \(-\sqrt{x}\) when the principal value is required). Assuming a partial function is total when the domain isn’t explicitly stated; forget to exclude points like \(x=3\) in \(f(x)=\frac{x^2+1}{x-3}\). Confusing preimage of an element with inverse function; the preimage may be a set, not a single value. Treating restriction as a new function with a different rule; it inherits the same rule, only on a smaller domain. ---