# Operations on function : addition, subtraction, multiplication

In this post we will try to understand how to perform arithmetic operations like addition, subtraction, multiplication and division on two given functions.

In order to keep things simple, i will help you understand these concepts with the help of examples as getting getting into mathematical technicalities can make things complex.

But before reading this post, i would strongly suggest you to read post about the basics of relation and functions. Its very important to have fair idea about what function is and how it is important for the point of view of mathematics.

## Arithmetic Operations on Function

Suppose you are provided with two functions f(x) and g(x)
The addition of these two functions can be represented as:

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

Hence the expression (f + g) (x) is representation of addition of function

You can see that the component of each function is simply added to get the desired result. You only need to have knowledge of simple arithmetic to perform this operation.

Let us understand this concept with the help of example:

Example 01
Let f(x) = x and g(x) = 2x + 1 be two real functions
Then find (f + g) (x)

Solution
As we have seen above the symbol (f + g) (x) means that the question is asking for addition of function.

In order to perform addition, perform a simple arithmetic operation on the function provided
(f + g) (x) => x + 2x + 1
(f + g) (x) => 3x + 1

Hence 3x + 1 is the right answer

Example 02
f (x) = 3{ x }^{ 2 }+5x+2
g (x) = 6{ x }^{ 2 }-2x-10
Find (f + g) (x)

Solution:
(f + g) (x) = 3{ x }^{ 2 }+5x+2 + 6{ x }^{ 2 }-2x-10
(f + g) (x) = 9{ x }^{ 2 }+3x-8

Example 03
f(x) = \sqrt { x }
g (x) = x
Find (f + g) (x)

Solution
(f + g) (x) = \sqrt { x } + x

### Subtraction of Function

Suppose you have been provided with two functions f(x) and g(x)

The subtraction between the two fraction can be represented as:

(f – g) (x) = f(x) – g(x)

The expression (f – g) (x) is the representation of subtraction of two functions

In this operation, the component of each function is subtracted with one another to get the desired result. This exercise is similar to the arithmetic operation of numbers, for this all we have to have is the knowledge of function.

Let us understand this concept with the help of example

Example 01
f (x) = 10x – 2
g (x) = 11x
Find (f – g) (x)

Solution
The expression (f – g) (x) means that the questions is asking to subtract function f (x) and g (x)
(f – g) (x) = 10x – 2 – 11x
(f – g) (x) = – x – 2
Hence (- x – 2) is the answer

Example 02
f (x) = { x }^{ 2 }
g (x) = 2x + 1
Find (f – g) (x)

Solution
The question is asking for subtraction of function f (x) – g (x)
(f – g) (x) = { x }^{ 2 }-2x-1
Hence { x }^{ 2 } - 2x - 1 is the right answer

Example 03
f (x) = { x }^{ 2 }-10x-2
g (x) = 5{ x }^{ 2 }-5x-5
Find (f – g) (x)

Solution
(f – g) (x) = { x }^{ 2 }-10x-2 5{ x }^{ 2 }+5x+5
(f – g) (x) = -4{ x }^{ 2 }-5x+3

### Multiplication of Function

In multiplication of function we basically perform simple arithmetic multiplication of two functions.

Suppose two functions are given f(x) and g(x)
The multiplication of the above function is given as:

(f · g) (x) = f(x) · g(x)

The expression for multiplication of function is (f · g) (x)

Let us see some examples for our understanding

Example 01
f (x) = 2x+1
g (x) = x
Find (f · g) (x)

Solution
The expression (f · g) (x) means that the question is asking for multiplication of two function

(f · g) (x) = (2x + 1) (x)
(f · g) (x) = 2{ x }^{ 2 }+\quad x

Example 2
f(x) = 2x
g(x) = \sqrt { x }
Find (f · g) (x)

Solution
(f · g) (x) = 2x . \sqrt { x } \\\ \\ \ (f · g) (x)=\quad 2.\quad x\quad .\quad x^{ \frac { 1 }{ 2 } }\\\ \\ (f · g) (x)=\quad 2.\quad x^{ \frac { 3 }{ 2 } }

### Division of Function

If f(x) and g(x) are the two function, then the division of function is given as

(f/g) (x) = f(x) / g(x)

The expression (f / g) (x) define that there is a division of two functions f(x) and g(x).
Let us understand the concept further with examples

Example 01
f(x) = { x }^{ 2 }
g(x) = { x }^{ 3 }
Find (f/g) (x)

Solution
The expression (f/g) (x) mean that the question is asking for division of function f(x) and g(x)

(f/g) (x) = \frac { { x }^{ 2 } }{ { x }^{ 3 } }
(f/g) (x) = \frac { 1 }{ { x } }

Hence 1/x is the answer for this question

Example 02
f(x) = 5x + 6
g(x) = { x }^{ 2 }
Find (f/g) (x)

Solution
(f/g) (x) = \frac { 5x\quad +\quad 6 }{ { x^{ 2 } } }

## Change in Domain after operation

We have already studied the concept domain in our previous post.
Now here we want to understand what would be the change of domain when we do arithmetic operation between two function.

Consider the following example
f (x) = x
g(x) = \sqrt { x }

The domain for f(x) can be represented as follows:
Domain of f(x)

All the numbers in the number line are part of the domain

Now let us find domain of function g(x)
Here the domain for \sqrt { x } is only positive number (including 0)

Domain of g(x)

We have removed negative number because square root of negative number is not possible.

Now if we do any Math operation on the function, the resulting domain will be the numbers which is common among both f(x) and g(x)

Suppose we perform f(x) + g(x)
(f + g) (x) = x + \sqrt { x }

The new domain will be the intersection of the domain elements of function f(x) and g(x)
Domain of f(x) + g(x)

Hence the domain for f(x) + g(x) is [0, infinity)

NOTES for DOMAIN
If we do any type of Math operation between the function, the resulting domain will be the common domain element between the two function

Example 02
f(x) = \sqrt { x }
g(x) = \sqrt { x-5 }
Find the domain for (f · g) (x)

Solution
Let us find domain of individual function first

f(x) = \sqrt { x }
Domain for f(x)

g(x) = \sqrt { x-5 }

The domain for \sqrt { x-5 } is [5, infinity)
Because if we take number less than 5, the equation gets negative and square root of negative number is not possible
Domain for g(x)

Then the domain for (f · g) (x) would be the common domain elements between f(x) and g(x)
Domain for (f · g) (x)

Hence the final domain of (f . g) (x) is [5, infinity)

Example 03 [Special case of division]

f(x) = \sqrt { x }
g(x) = \sqrt { x-5 }
Find domain of (f/g) (x)

Solution
Domain of f(x) = \sqrt { x } is:

Domain of g(x) =\sqrt { x-5 } is

The function of (f/g) (x) is written as \frac { \sqrt { x } }{ \sqrt { x-5 } }

We know that the domain of operation is equal to common domain elements of both the functions, but in this case we have to include extra scenario of denominator getting zero.

If the value of x such that the denominator of fraction gets zero, then the result will be infinity which is not a mathematical solution.

For example if we put (x=5) in \frac { \sqrt { x } }{ \sqrt { x-5 } } , the result will be infinity.
So we also have to remove number 5 from the domain.

Hence the final domain of function (f/g) (x) is