# 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**

**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.

**Representation of 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 \(D=[1, 2, 3]\), \(R=[1,4, 9]\).

Here is a representation of a function in math as an ordered pair:

**Identification of a function in math**

Function in mathematics means correspondence from one value \(x\) of the first set to another value \(y\) of the second set. This correspondence can be of the following four types. But not every correspondence is a function.

In the following example, only \(1 – 1\) and many to one are examples of a function because no two ordered pairs have the same first component and all elements of the first set are related to them. So we say that in a function one input can result in only one output. If every \(x\) is given to us, then there is one and only one \(y\) that can be paired with that \(x\). A function cannot be connected to two outputs.

The curve drawn in the diagram is shown as a function, then each vertical line intersects the curve at a maximum of one point. This vertical line test helps us determine whether a curve is a function or not.

**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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