How to Determine Functions?

The functions are the special types of relations. A relation is a function in which each element of set \(A\) has one and only one image in set \(B\). The function is represented as \(f: A → B\).

Related Topics

• How to Determine a Function Using the Vertical Line Test?

A step-by-step guide to determining functions

A function is a process or a relation that associates each element \(a\) of a non-empty set \(A\), at least to a single element \(b\) of another non-empty set \(B\). The relation \(f\) from a set \(A\) (function domain) to another set \(B\) (the co-domain of the function) is called a function in math. \(f = {(a,b)| for\:all\:a ∈ A, b ∈ B}\)

• A relation is a function in which each element of set \(A\) has one and only one image in set \(B\).
• A function is a relation of a non-empty set \(B\) such that the domain of a function is \(A\) and no two distinctly ordered pairs in \(f\) have the same first element.
• A function from \(A → B\) and \((a,b) ∈ f\), then \(f(a) = b\), where \(b\) is the image of \(a\) under \(f\) and \(a\) is the preimage of \(b\) under \(f\).
• If there exists a function \(f: A → B\), set \(A\) is called the domain of the function \(f\), and set \(B\) is called its co-domain.

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How to represent functions in math?

The rule that defines a function can take many forms, depending on how it is defined. They can be defined as piecewise-defined functions or as formulas.

A function in math can be represented as:

• a set of ordered pairs
• an arrow diagram
• a table form
• a graphical form

\(f (x) = x^2\) is the general way to display a function. It is said as \(f\) of \(x\) is equal to \(x\) square. This is shown as \(f = [(1,1), (2,4), (3,9)]\). The domain and range of a function is given as \(D=[1, 2, 3]\), \(R=[1,4, 9]\).

Types of functions

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Functions are very important in mathematics and let us study different types of functions. We have four functions based on the mapping of elements from set \(A\) to set \(B\).

1. \(f: A → B\) is called one-to-one or injection, if the images of distinct elements \(A\) under \(f\) are distinct, that is, for each \(a, b\) in \(A, f (a) = f (b), ⇒ a = b\). Otherwise, a few to one.
2. \(f: A → B\) is onto if each element \(B\) is an image of some element of \(A\) under \(f\), that is, for every \(b ϵ B\), there is an element \(a\) in \(A\) such that \(f (a) = b\). A function is onto if and only if the function range is \(= B\).
3. \(f: A → B\) is one-to-one and onto or bijective if \(f\) is both one-one and onto.

Determining Functions – Example 1:

Is this a function? \([(−2,2),(−3,3),(−4,4)]\)

Solution:

In this set of ordered pairs, the domain values are \(−2, −3, −4\) and the range values are \(2,3,4\). Since each domain value is paired with only one range value, this relationship is a function.

Exercises for Determining Functions

Which set of ordered pairs shows a relationship that is a function?

1. \(\color{blue}{(1,−3),(2,−4),(0,3),(−1,6),(−2,9)}\)
2. \(\color{blue}{(−5,−5),(−6,−6),(−5,−7),(−4,−2),(−5,−7)}\)
3. \(\color{blue}{(2,1),(2,2),(2,3),(2,4),(2,5)}\)

1. \(\color{blue}{Yes}\)
2. \(\color{blue}{No}\)
3. \(\color{blue}{No}\)