Determine the image of a function when the object is given and vice versa

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)

The diagram shows an illustration of two sets, namely set x and set y, connected by a one-to-one relationship

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

Graph displaying a point and line, illustrating a one-to-one function with the vertical line test applied.

Not a function

A non-function graph with a circle intersecting a vertical line, function for example does not pass the vertical line test.

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:

Image of two unique numbers on square, showcasing one-to-one mapping in function f(x) = x^2.

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

Graph displaying a point and line, illustrating a one-to-one function with the vertical line test applied.

Not a function

A non-function graph with a circle intersecting a vertical line, function for example does not pass the vertical line test.

Domain and Range

Example

Question

Given an arrow diagram of a function:

Image of two unique numbers on square, showcasing one-to-one mapping in function f(x) = x^2.

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

Functions

Chapter : Functions

Topic : Determine the image of a function when the object is given and vice versa

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