## Functions

 1.1 Functions

 Definition A function from set $$X$$ to set $$Y$$ is a special relation that maps each element $$x$$ in set $$X$$ to only one element $$y$$ in set $$Y$$.

 Example of Function (One-to-one)

 Function Notation Any element $$x$$ in set $$X$$ that is mapped to one element $$y$$ in set $$Y$$ by $$y=2x+1$$ is written in function notation as below: $$f:x\rightarrow y$$ or $$f(x)=y$$ $$f:x\rightarrow 2x+1$$ or $$f(x)=2x+1$$ where $$x$$ is the object and $$2x+1$$ is the image.

Graph
 Function

 Not a function

 Domain and Range In general, the domain of a function is the set of possible values of $$x$$ which defines a function, whereas range is the set of values of $$y$$ that are obtained by substituting all the possible values of $$x$$.

Example
 Question

Given an arrow diagram of a function:

Find the domain, codomain and range for the function, hence

state the object of $$4$$ and image of $$3$$

 Solution

From the diagram above,

Domain $$=\lbrace1,\,2, \, 3,\,5 \rbrace$$,
Codomain $$=\lbrace1,\,4, \, 9,\,16,\,25 \rbrace$$,
Range $$=\lbrace1,\,4, \, 9,\,25 \rbrace$$.

Next,

Object of $$4$$ is $$2$$ and image of $$3$$ is $$9$$.

## Functions

 1.1 Functions

 Definition A function from set $$X$$ to set $$Y$$ is a special relation that maps each element $$x$$ in set $$X$$ to only one element $$y$$ in set $$Y$$.

 Example of Function (One-to-one)

 Function Notation Any element $$x$$ in set $$X$$ that is mapped to one element $$y$$ in set $$Y$$ by $$y=2x+1$$ is written in function notation as below: $$f:x\rightarrow y$$ or $$f(x)=y$$ $$f:x\rightarrow 2x+1$$ or $$f(x)=2x+1$$ where $$x$$ is the object and $$2x+1$$ is the image.

Graph
 Function

 Not a function

 Domain and Range In general, the domain of a function is the set of possible values of $$x$$ which defines a function, whereas range is the set of values of $$y$$ that are obtained by substituting all the possible values of $$x$$.

Example
 Question

Given an arrow diagram of a function:

Find the domain, codomain and range for the function, hence

state the object of $$4$$ and image of $$3$$

 Solution

From the diagram above,

Domain $$=\lbrace1,\,2, \, 3,\,5 \rbrace$$,
Codomain $$=\lbrace1,\,4, \, 9,\,16,\,25 \rbrace$$,
Range $$=\lbrace1,\,4, \, 9,\,25 \rbrace$$.

Next,

Object of $$4$$ is $$2$$ and image of $$3$$ is $$9$$.