Operations on Functions: Addition, Subtraction, Multiplication & Division Explained

Function is an expression, rule, or law that defines a relationship between one variable and another variable. A function is written as , which is read as “f of x” or function of x, and y and x are related such that for every x, there is a unique value of y. There are four main operations that can be performed on functions: addition, subtraction, multiplication and division.

Function Composition: A function has 2 sets – domain set and range set. The set of values that ‘x’ can take upon such that the function is well-defined is the domain set whereas the set of all values obtained for ‘y’ is called the range set.

Example: , is a function. Here the value of ‘x’ can be 3 or 4 or 0 or -6 or any real number in general, these all values of ‘x’ comprises the domain set, and with these values of ‘x’ we get the values of ‘y’ as 3 or 5 or -3 or -15 or other real numbers respectively, all these values of ‘y’ comprises the range set.

What are Operations on Functions?

There are mainly 4 types of operations on functions in maths that take place between any two functions:

1. Addition operation of Functions
2. Subtraction operation of Functions
3. Multiplication operation of Functions
4. Division operation of Functions

Remember only functions with overlapping domains can be added, subtracted, multiplied and divided.

Let us see these operations on functions with some examples.

If we consider two functions and then for all x belonging to the domain of both the functions the addition operation on functions is defined as ,

Thus, the value of the sum of the two functions at an argument is equal to the sum of the values of the functions taken individually at the argument.

For example if we have, and ,then for x=2, we get

Or alternatively,

Thus the result of addition of these two functions for x=2 is 28.

Subtraction Operation of Functions

If we consider two functions and then for all x belonging to the domain of both the functions the subtraction operation on functions is defined as,

Thus, the value of the difference of the two functions at an argument is equal to the difference of the values of the functions taken individually at the argument.

Let us take two functions, and , then for x=3, we get

Alternatively,

Thus the result of subtraction of these two functions is 2.

Multiplication Operation of Functions

Let us consider two functions and then for all x belonging to the domain of both the functions the multiplication operation on functions is defined as,

Thus, the value of the product of the two functions at an argument is equal to the product of the values of the functions taken individually at the argument.

For example we are given two functions such as, and , then using the formula for x=0, we get

Alternatively,

Thus the result of multiplication of these two functions is 0.

Division Operation of Functions

If we are given two functions, and then for all x belonging to the domain of both the functions the division operation on functions is defined as,

, where .

Thus, the value of the division of the two functions at an argument is equal to the division of the values of the functions taken individually at the argument.

For example, and , then for x=3, we

Thus the result for division of these two functions is 3.

Let us now see a few solved examples to understand better.

Composition of Functions

Composition of functions refers to combining two or more functions to create a new function. It involves using the output of one function as the input for another function. The function whose value at x is f(g(x)) is called the composite of the functions f
and g.

For example, let's consider two functions:
f(x) and g(x). The composition of f and g, denoted as (f∘g)(x), means applying
f to the result of g(x).

To illustrate this, suppose we have
f(x)=2x and g(x)=x+3. To find (f∘g)(x), we first evaluate g(x), which is
x+3, and then substitute this result into f(x). So,
(f∘g)(x)=f(g(x))=f(x+3)=2(x+3)=2x+6.

In this case, the composition of
f and g creates a new function, 2x+6, where the output of g(x) becomes the input for
f(x).

Composition of functions allows us to combine different operations and simplify calculations.

Operations on Functions Solved Examples

Example 1: If and , find the value of

Solution: We are provided with two functions, and .

We know the addition operation on functions is defined as, .

Here we are also given the value of ‘x’ i.e. 2.

Therefore

Thus the result of addition of these two functions is 4.

Example 2: If and , find the value of

Solution: We know that the subtraction operation on functions is defined as,

Therefore with the given value of ‘x’ i.e., 1, we get

Thus the result of subtraction of these two functions is -3.

Example 3: and , then find the value

Solution: We know that the multiplication operation on function is defined as,

We are also given the value of ‘x’ as -2.

So,

Thus the result of multiplication of these two functions is 9.

Example 4: If and , then find

Solution: We know that the division operation on function is defined as,

,

But we have to check if the denominator is non-zero or not.

Therefore, .

As, , therefore we can proceed with the formula.

Therefore, .

Thus the result of division of these two functions is .

If you are checking Operations on Functions article, also check the related maths articles:
Cosine Function	Domain and Range of a Function
Types of Functions	Periodic Function
Many One Function	Logarithmic Functions

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