Functions¶
Recall (doc/postdoc). Not a first course.
Job: morphisms of sets \(f:A\to B\); later continuous / measurable / smooth structure on top.
Core¶
\(f:A\to B\) assigns each \(x\in A\) exactly one \(f(x)\in B\).
| Term | Definition |
|---|---|
| Domain / codomain | \(A\) / \(B\) |
| Image | \(f(A)=\{f(x):x\in A\}\) |
| Injective | \(f(x)=f(y)\Rightarrow x=y\) |
| Surjective | \(f(A)=B\) |
| Bijective | inj + surj \(\Leftrightarrow\) inverse \(f^{-1}:B\to A\) |
| Composition | \((g\circ f)(x)=g(f(x))\) |
Graph: \(\Gamma_f\subset A\times B\). Over \(\mathbb{R}\): vertical line test for graphs of functions \(y=f(x)\).
Real domain exclusions (typical): denom \(=0\); even root of negative; \(\log\) of nonpositive — describe as preimages, not folklore.
Used later for¶
- Exp & log: inverse pair on restricted domains
- Category theory: \(\mathbf{Set}\) morphisms
- Topology: continuous maps
- Measure theory: measurable maps, pushforwards
Check¶
Is \(f:\mathbb{R}\to\mathbb{R}_{\ge0}\), \(f(x)=x^2\) bijective?
Surjective onto \(\mathbb{R}_{\ge0}\) yes; injective no (\(f(1)=f(-1)\)). Bijective if domain restricted to \([0,\infty)\) with same codomain.