# How to Determine Functions?

A function in mathematics is represented as a rule, which gives a unique output for each input \(x\). In this step-by-step guide, you will learn more information about defining functions and how to identify them.

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**

**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.

**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**

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 \(B\).

- \(f: A → B\) 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.
- \(f: A → B\) 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 \(= B\).
- \(f: A → B\) 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?**

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

- \(\color{blue}{Yes}\)
- \(\color{blue}{No}\)
- \(\color{blue}{No}\)

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