**Introduction**

The concept of function is of paramount importance in mathematics and among other disciplines as well. Functions form one of the most important building blocks of Mathematics. The word function has been derived from the Latin word meaning operation and the words mapping and map are synonymous to it. Functions play an important role in differential and integral calculus. Before moving to study addition and subtraction of functions, let us first understand what we mean by the term function.

**Function as a Special Kind of Relation**

Let A and B be two non-empty sets. A relation f from A to B, i.e. a subset of A x B is called a function or a mapping or a map from A to B is,

- For each a A there exists b B such that ( a, b ) ∈ f
- ( a, b ) ∈ f and ( a, c ) ∈ f ⇒ b = c

Thus a non-void set f of A x B is a function from A to B if each element of A appears in some ordered pair in f and no two ordered pairs in f have the same first element.

If ( a, b ) ∈ f, then b is called the image of a under f.

**Domain, Co-domain and Range of a Function**

Let f : A → B. then the set A is known as the domain of f and the set B is known as the range co-domain of f. The set of all f-images of elements of A is known as the range of f or image set of A under f and is denoted by f ( A ).

Thus f ( A ) = { f (x) : x ∈ A } = Range of f

**Real Functions**

Some functions have domain and co-domain both as subsets of the set R of all real numbers. Such functions are called real functions or real valued functions of the real variable. In other words, **a function f : A → B is called a real valued function, if B is a subset of R, where R is the set of all real numbers.**

Let us now discuss two major operations on real function, i.e. addition and subtraction. Is it similar to the normal addition of real numbers or there are different rules for addition and subtraction of real functions Let us find out.

**Addition of Functions**

Let f : D_{1} → R and f : D_{2} → R bet two real functions, then, their sum f + g is defined as the function D_{1} ∩ D_{2} to R which associates each x ∈ D_{1} ∩ D_{2} to the number f ( x ) + g ( x ).

In other words, f : D_{1} → R and f : D_{2} → R bet two real functions, then their sum f + g is a function from D_{1} ∩ D_{2} to R such that

( f + g ) ( x ) = f ( x ) + g ( x ) for all x ∈ D_{1} ∩ D_{2}.

Let us understand it by an example.

**Example**

Find the sum of the identity function and the modulus function.

**Solution**

We know that f : R → R defined by f ( x ) = x is the identity function and f : R → R defined by g ( x ) = | x | is the modulus function. Clearly, f and g have the same domain.

Therefore, f + g → R

Now, ( f + g ) ( x ) = f ( x ) + g ( x )

⇒ ( f + g ) ( x ) = x + | x |

⇒ ( f + g ) ( x ) = { x+x , if x>0 x-x, if x<0

⇒ ( f + g ) ( x ) = { 2x , if x>0 0, if x<0

**Thus f → g : R → R is defined as **

**( f + g ) ( x ) = **{ 2x , if x>0 0, if x<0

Let us now understand how to find the difference of functions.

There are two methods of adding the functions

**Horizontal Method**

To add functions using this method, arrange the functions added in a horizontal line and collect all the groups of like terms and then add.

Let us understand this by an example.

**Example**

Add f ( x ) = x + 2 and g ( x ) = 5x – 6

**Solution**

We have been given the functions, f ( x ) = x + 2 and g ( x ) = 5x – 6

( f + g ) ( x ) = f ( x ) + g ( x )

= ( x + 2 ) + ( 5x – 6 )

= 6x – 4

**Hence, ( f + g ) ( x ) =6x – 4**

**Vertical Method**

In this method, the elements of the functions are arranged in columns and then added.

Let us understand this by an example.

**Example**

Add f ( x ) = x + 2 and g ( x ) = 5x – 6

**Solution**

We have been given the functions, f ( x ) = x + 2 and g ( x ) = 5x – 6

( f + g ) ( x ) = f ( x )

+ = g ( x )

——————————

——————————

Therefore, we have,

x + 2

+5x – 6

———-

6x – 4

———-

**Hence, ( f + g ) ( x ) =6x – 4**

**Subtraction of Functions**

Let f : D_{1} → R and f : D_{2} → R bet two real functions, then, their difference of g from f, denoted by f – g is defined as the function D_{1} ∩ D_{2} to R which associates each x D_{1} ∩ D_{2} to the number f ( x ) – g ( x ).

In other words, f : D_{1} → R and f : D_{2} → R bet two real functions, then their difference of g from f, denoted by f – g is a function from D_{1} ∩ D_{2} to R such that

( f – g ) ( x ) = f ( x ) – g ( x ) for all x ∈ D_{1} ∩ D_{2}.

Let us understand it through an example.

**Example**

Find the difference of the identity function and the modulus function.

**Solution**

We know that f : R → R defined by f ( x ) = x is the identity function and f : R → R defined by g ( x ) = | x | is the modulus function. Clearly, f and g have the same domain.

Therefore, f – g → R

Now, ( f – g ) ( x ) = f ( x ) – g ( x )

⇒ ( f – g ) ( x ) = x – | x |

⇒ ( f – g ) ( x ) = { x-x , if x>0 x-( – x ), if x<0

⇒ ( f – g ) ( x ) = { 0 , if x>0 2x, if x<0

**Thus f → g : R → R is defined as **

**( f – g ) ( x ) = **{ 0 , if x>0 2x, if x<0

Let us now understand how to find the difference of functions.

There are two methods of subtracting the functions

**Horizontal Method**

To subtract functions using this method, arrange the functions to be subtracted in a horizontal line and collect all the groups of like terms and then subtract.

Let us understand this by an example.

**Example**

Subtract g ( x ) = 5x – 6 from f ( x ) = x + 2

**Solution**

We have been given the functions, f ( x ) = x + 2 and g ( x ) = 5x – 6

( f – g ) ( x ) = f ( x ) – g ( x )

= ( x + 2 ) – ( 5x – 6 )

= x + 2 -5x + 6

= – 4x + 8

**Hence, ( f – g ) ( x ) = – 4x + 8**

**Vertical Method**

In this method, the elements of the functions are arranged in columns and then subtracted.

Let us understand this by an example.

**Example**

Add f ( x ) = x + 2 and g ( x ) = 5x – 6

**Solution**

We have been given the functions, f ( x ) = x + 2 and g ( x ) = 5x – 6

( f – g ) ( x ) = f ( x )

– g ( x )

————————

————————

Therefore, we have,

x + 2

– 5x – 6

———-

– 4x + 8

———-

**Hence, ( f – g ) ( x ) = – 4x + 8**

**Graphical representation of Addition and Subtraction of Functions**

Let us now see how to plot the graph as well their sum or difference. Let us understand this through an example.

**Graphical Representation of Addition of Functions **

Let f ( x ) = 2x + 3 and g ( x ) = x^{2}

f ( x ) + g ( x ) = 2x + 3 + x^{2}

( f + g ) ( x )= x^{2} + 2x + 3

The graphs of these two functions and their addition will be represented as

**Graphical Representation of Subtraction of Functions **

Let f ( x ) = 2x + 3 and g ( x ) = x^{2}

f ( x ) – g ( x ) = 2x + 3 – x^{2}

( f – g ) ( x )= – x^{2} + 2x + 3

The graphs of these two functions and their subtraction will be represented as

**Properties of Addition and Subtraction of Functions**

Let us now check whether the addition and subtraction of functions satisfy the basic properties of operations.

**Commutative Property**

Commutative Property states that when an operation is performed on two numbers, the order in which the numbers are placed does not matter. Let us check whether the addition and subtraction of functions satisfy the communicative property or not.

For this purpose, let us consider two functions, f ( x ) = 3x^{2} – 4x + 8 and g ( x ) = 5x + 6

Let us first find f ( x ) + g ( x ) = 3x^{2} – 4x + 8 + 5x + 6

⇒ f ( x ) + g ( x ) = 3x^{2} + x + 14 …………………. ( 1 )

Now let us find, g ( x ) + f ( x ) = 5x + 6 + 3x^{2} – 4x + 8

⇒ g ( x ) + f ( x ) = 3x^{2} + x + 14 …………………. ( 2 )

From ( 1 ) and ( 2 ), we can say that f ( x ) + g ( x ) = g ( x ) + f ( x )

**Hence, addition of functions is commutative. **

Now, let us check whether subtraction of functions is commutative.

Again, consider the functions, f ( x ) = 3x^{2} – 4x + 8 and g ( x ) = 5x + 6

Let us first find f ( x ) – g ( x ) = 3x^{2} – 4x + 8 – ( 5x + 6 )

⇒ f ( x ) – g ( x ) = 3x^{2} – 4x + 8 – 5x – 6

⇒ f ( x ) – g ( x ) = 3x^{2} – 9x + 2 …………………. ( 1 )

Now let us find, g ( x ) – f ( x ) = 5x + 6 – ( 3x^{2} – 4x + 8 )

⇒ g ( x ) – f ( x ) = 5x + 6 – 3x^{2} + 4x – 8

⇒ g ( x ) – f ( x ) = – 3x^{2} + + 9x – 2 …………………. ( 2 )

From ( 1 ) and ( 2 ) we can say that f ( x ) – g ( x ) g ( x ) – f ( x )

**Hence, subtraction of functions is not commutative.**

**Associative Property**

Associative Property states that when an operation is performed on more than two numbers, the order in which the numbers are placed does not matter. Let us check whether the addition and subtraction of functions satisfy the associative property or not.

For this purpose, let us consider three functions, f ( x ) = 4x + 8 and g ( x ) = 5x + 6 and h ( x ) = x + 1

Let us first find f ( x ) + [ g ( x ) + h ( x ) ] = 4x + 8 + [ ( 5x + 6 ) + ( x + 1 ) ]

⇒ f ( x ) + [ g ( x ) + h ( x ) ] = 4x + 8 + [ 6x + 7 ]

⇒ f ( x ) + [ g ( x ) + h ( x ) ] = 10x + 15 …………………. ( 1 )

Now, let us calculate [ f ( x ) + g ( x ) ] + h ( x ) = [ (4x + 8) + ( 5x + 6 ) ] + ( x + 1 )

⇒ [ f ( x ) + g ( x ) ] + h ( x ) = [ 9x + 14 ] + ( x + 1 )

⇒ [ f ( x ) + g ( x ) ] + h ( x ) = 10x +15 …………………. ( 1 )

From ( 1 ) and ( 2 ), we can say that f ( x ) + [ g ( x ) + h ( x ) ] = [ f ( x ) + g ( x ) ] + h ( x )

**Hence addition of functions is associative.**

Since, subtraction of functions is not commutative; therefore, it will not be associative as well.

**Key Facts and Summary**

- Let A and B be two non-empty sets. A relation f from A to B, i.e. a subset of A x B is called a function or a mapping or a map from A to B is,

- For each a A there exists b B such that ( a, b ) ∈ f
- ( a, b ) ∈ f and ( a, c ) ∈ f ⇒ b = c

- Let f : A → B. then the set A is known as the domain of f and the set B is known as the range co-domain of f. The set of all f-images of elements of A is known as the range of f or image set of A under f and is denoted by f ( A ).
- A function A → B is said to be onto function or a surjection if every element of B is the f-image of some element of A, i.e. f ( A) = B or range of f is the c-domain of f.
- A function A → B is said to be a one-one function or an injection if different elements of A have different images of B.
- A function A → B is said to be a many-one function if two or more elements of set A have the same image in B.
- A function f : A → B is called a real valued function, if B is a subset of R, where R is the set of all real numbers.
- Let f : D
_{1}→ R and f : D_{2}→ R bet two real functions, then, their sum f + g is defined as the function D_{1}∩ D_{2}to R which associates each x ∈ D_{1}∩ D_{2}to the number f( x ) + g( x ). - Let f : D
_{1}→ R and f : D_{2}→ R bet two real functions, then, their difference of g from f, denoted by f – g is defined as the function D_{1}∩ D_{2}to R which associates each x ∈ D_{1}∩ D_{2}to the number f ( x ) – g ( x ).

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