# 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 $$f: A → B$$.

## Step by step guide todetermining 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 $$B$$.

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. $$\color{blue}{No}$$

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