BCS1130-8 Functions

Lecture notes

• Definition of a function - Examples of function definition
• Composition of functions - Examples

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Definition of a function

Definition

A function f is a mapping from a set A to a set B. To be a function, $f(x)$ has a unique value for every element x of A: $f:A\to B,f(x)=\dots$

• A → Domain (asse x)
• B → Codomain (asse y)
• Range: $\{f(x),x\in A\},range(f)\subseteq B$

• Codomain and Range are different!

Examples of function definition

• $f:N\to R$ where $f(x)=\sqrt{(}x)$
• $f:N\to R$ where $f(x)=\frac{1}{x}$

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Composition of functions

Definition

Consider functions $g:A\to B$ and $f:B\to C$ Then $(f\circ g)(x)=f(g(x))$ is the function obtained first using g and then using f

• Domain of $(f\circ g)$ → domain of g
• Codomain of $(f\circ g)$ → codomain of f

• This condition must be true for the compound function: $range(g)\subseteq domain(f)$

Examples

• $f:Z\to Z,f(x)=x-4$

• $g:Z\to Z$, where $g(x)=x^2+4$

• $f\circ g:Z\to Z$ is properly defined, since the codomain of g and the domain of f are the same

• $g\circ g:Z\to Z$ is properly defined, since the codomain of f and the domain of g are the same

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Key takeaways

• Do you understand what a function is?
• Do you know the meaning of ‘domain’, ‘co-domain’ and ‘range?
• Do you understand how function composition works?

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Lecture slides