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Introduzione e definizione formale; Dominio, codominio e immagine; Funzioni iniettive, suriettive e biiettive; Funzione inversa e composizione; Applicazioni e discussione

Le Funzioni

Matematica

Le Funzioni

Comprendere le funzioni: basi, tipi e applicazioni.

Indice

• Introduzione e definizione formale

• Dominio, codominio e immagine

• Funzioni iniettive, suriettive e biiettive

• Funzione inversa e composizione

• Applicazioni e discussione

Le Funzioni: Introduzione

Le funzioni modellano la dipendenza di una variabile da un'altra, come la traiettoria di un proiettile o la crescita demografica. Sono strumenti essenziali in fisica, ingegneria, economia e data science per analizzare e prevedere fenomeni complessi.

Definizione Formale

Una funzione f: A → B è una relazione che associa ad ogni elemento x ∈ A uno e un solo elemento y ∈ B, denotato y = f(x). Ad esempio, f(x) = x² associa ogni numero reale x al suo quadrato. Qui A è il dominio e B è il codominio.

Funzioni: dominio e codominio, con esempi.

Funzione iniettiva: Mappa dominio a codominio distinti.

Dominio e Codominio

• **Dominio (D_f)**: Insieme dei valori 'x' per cui f(x) è definita.

• Es. `f(x)=1/x` implica `D_f = R \ {0}`.

• **Codominio (C_f)**: Insieme di arrivo della funzione (target set).

• Non sempre coincide con l'immagine (range) della funzione.

Immagine di un Elemento

L'immagine di un elemento x tramite una funzione f è il valore f(x) ottenuto applicando la regola funzionale. Rappresenta l'output 'y' corrispondente all'input 'x'.

Controimmagine di un Valore

La controimmagine di un valore y è l'insieme degli x nel dominio per cui f(x) = y. Per f(x) = x² + 3, la controimmagine di y=7 si trova risolvendo x² + 3 = 7, ottenendo x = ±2.

Funzioni algebriche: mappatura di input a output.

Insieme Immagine

L'Insieme Immagine (Im(f)) è il sottoinsieme del codominio con i valori effettivamente assunti dalla funzione. Per f(x)=x², Im(f)=[0,+∞), mentre il codominio può essere R. Si determina graficamente o analiticamente.

Funzioni: dominio e codominio, f(x)=x².

Funzioni Reali Reali

Le funzioni reali di variabile reale mappano un sottoinsieme di ℝ a ℝ (f: D ⊆ ℝ → ℝ). La loro rappresentazione grafica nel piano cartesiano può essere continua (es. y = x²) o discreta (es. y = ⌊x⌋).

Grafico di una funzione quadratica, essenziale per capire le funzioni reali.

Funzioni Iniettive

• Definizione: `f(x₁) = f(x₂) ⇒ x₁ = x₂` (elementi distinti hanno immagini distinte).

• Ogni immagine nel codominio proviene da al più un elemento del dominio.

• Test grafico: ogni linea orizzontale interseca il grafico al più in un punto.

• Esempi: `f(x) = x³` è iniettiva; `f(x) = x²` non lo è.

Le Funzioni

Introduzione e definizione formale; Dominio, codominio e immagine; Funzioni iniettive, suriettive e biiettive; Funzione inversa e composizione; Applicazioni e discussione

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