Functions are a tool to describe the real world in mathematical terms.

> The area of a circle depends on the radius of the circle.

Area of the circle

> The distance an object travels at constant speed along a straight line path depends on the elapsed time.

Distance travelled by an object at a constant speed

In both the cases mentioned above, the value of one variable quantity, say \boldsymbol{y}, depends on the value of another variable quantity, which we might call \boldsymbol{x}. We say that \boldsymbol{y} is a function of \boldsymbol{x} and write this symbolically as:

\boldsymbol{y = f(x)}

The symbol \boldsymbol{‘f’} represents the function. The letter \boldsymbol{‘x’} is the independent variable representing the input value of the function. \boldsymbol{‘y’} is the dependent variable or output value of \boldsymbol{f} at \boldsymbol{x}.

function representation

Suppose we have two sets ‘D’ and ‘Y’ such that every element in the set D is related to exactly one element in the set Y. Such a relation is called a function.

Set D and Set Y

A function \boldsymbol{f} from a set D to a set Y is a rule that assigns exactly one element from the set Y to each element in the set D.

> The set D of all possible input values is called the domain of the function.

> The set Y is called the codomain of the function.

> The set of all output values of f(x) as x varies throughout the set D is called the range of the function.

In this example, Range = {C, D, E}.

Domain, Codomain and Range in Set D and Set Y

Note that the range may not include every element in the set Y. The range is the subset of the codomain. It may or may not be equal to the codomain.

Range \boldsymbol{\subseteq} Codomain

There are two important rules for a relation between input and output sets to be a function:

1. Every element in the domain must have a corresponding value in the codomain. In other words, every element in the input set must be related to some element in the output set.

2. A function is single-valued. It can’t give two or more output values for the same input value. In other words, every element in the input set must be related to exactly one element in the output set.

A function f is like a machine that produces an output value f(x) in its range whenever we feed it an input value x from its domain.

function and machine analogy

Example 1: Consider the sets D and Y related to each other as shown below. Can we consider this relation as a function?

Functions Example 1 Set D and Set Y

Solution:

Every element in the set D is related to exactly one element in the set Y. So, the given relation is a function.

Note that in this case, the range is same as the codomain.

Range = Codomain = {A, B, C, D}

Example 2: Can we consider the relation shown below as a function?

Functions Example 2 Set D and Set Y

Solution:

Every element in the set D is related to exactly one element in the set Y. So, the given relation is a function.

In this case, the range is not equal to the codomain.

Range = {B, C, D}

Codomain = {A, B, C, D}

Note that a function can have same output value at two or more input elements in the domain but it should not have more than one output value at the same input element in the domain.

Example 3: Can we consider the relation shown below as a function?

Functions Example 3 Set D and Set Y

Solution:

Every element in the set D is related to some element in the set Y. But every element in the set D is not related to exactly one element in the set Y. The element 2 in the input set is assigned to two elements, B and C, in the output set. So, the given relation is not a function.

Example 4: Can we consider the relation shown below as a function?

Functions Example 4 Set D and Set Y

Solution:

The element 4 in the input set is not assigned to any element in the output set. So, the given relation is not a function.

Example 5: Can we consider the relation shown below as a function?

Functions Example 5 Set D and Set Y

Solution:

The element 4 in the input set is not assigned to any element in the output set. Also, the element 2 in the input set is assigned to two elements, B and C, in the output set. So, the given relation is not a function.

Vertical Line Test for a Function:

Not every curve in the coordinate plane can be the graph of a function. Using this test, we can figure out whether a given curve is the graph of a function or not.

As discussed earlier, a function f can have only one value f(x) for each x in its domain. So, no vertical line can intersect the graph of a function more than once.

If a is in the domain of the function f, then the vertical line x=a will intersect the graph of f at the single point (a,f(a)).

Consider the curve in the coordinate plane as shown below:

Curve in a coordinate plane

A vertical line intersects the above curve twice. So, the above curve is not the graph of a function.

Similarly, the circle is not the graph of a function, since a vertical line can intersect the circle twice.

However, the upper semicircle and the lower semicircle both are the graphs of functions.

Upper semicircle and lower semicircle

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What is a function in mathematics?
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