Sum of functions

Introduction

A rule that assigns each element in X to exactly one element in Y is called a function. If a variable y is associated with an independent variable x that there is a rule that determines a unique value of y whenever a numerical value is assigned to x, then y is said to be a function of the independent variable x.

This relationship is often denoted by the notation y = f(x), which means “f of x,” and y and x are coupled in such a way that there is a unique value of y for each x. That is, for the same x, f(x) cannot have more than one value.

Sum Function of Real Numbers

Methods for calculating Sum of Function 

Two methods can be implied to calculate the sum of function. You may either add the functions together and then input the value for x and simplify, or you can add the functions together and then put the value for x and simplify.

The sum of functions for the term is  written as:

(a+b)(x) = a(x)+b(x)

Finding Sum of Function

Let’s take a complex number (a+b)(3) , where a(x) = x2 – x + 4 and b(x) = x-2.

Find (a+b ) 3

Method 1:

We can calculate the sum of functions by multiplying each term with 3 and then solving separately.

(a+b)(3) can be written as  (a)(3)+b(3), or 

(a+b)(3) = (a)(3)+b(3)

Putting the value of x=3 into each function and then adding the results together.

Step 1: Calculate value of a(3)

a(x) = x2 – x + 4

a(3)=(3)2-3+4

a(3)=(3)​2 −3+4

Applying simplifying rule 

a(3)=9-3+4

a(3)=9−3+4

a(3)=6+4

a(3)=10

Step 2 :Calculate the value of b(3)

 b(x) = x-2, putting the value of x=3, we get 

b(3)=3-2

b(3)=3−2

b(3)=1

Step 3: Calculate the sum of functions in the original equation (a+b)(3) = (a)(3)+b(3) can be calculated by inputting the value of (a)3 and b(3).

 (a+b)(3) = (a)(3)+b(3)

(a+b)(3) = 10+1

(a+b)(3) =11

Method 2

First add the functions together and then input the value of x=3 to calculate the sum of the function

(a+b)3 , where a(x) = x2 – x + 4 and b(x) = x-2.

Step 1:  Put the value of a and b in the equation

(a+b)(x)= x2 – x + 4 + (x-2)

Simplify and add the  like terms.

(a+b)(x) =  x2 – x + 4 + x – 2

(a+b)(x) =  x2 +2

Step 2: Put the value of x = 3 in above equation

(a+b)(x) =  x2 +2

(a+b)(3) =  (3)2 + 2

(a+b)(3)=9+2

(a+b)(3)=11

Sum of Functions for Complex Numbers

What is a complex number?

A complex number is a number expressed in the form of a + bi, where a or b or both may be equal to zero. The imaginary number is i while the real numbers are a and b.

  i  =-1

 Thus, complex numbers  consist of many different parts. 

Addition Rule of a complex number

When performing arithmetic operations on complex numbers, similar terms should be added. The solution must be written in the simplest a+ bi form.

For example ADD: (8+6i) + (10 – 4i)

Express answer in a + bi form.

(8+6i) + (10 – 4i)= 8 + 6i +10 – 4i = 18 + 2i

Or by the grouping rule:

(8+6i) + (10 – 4i) = (8+10) + (6-4) i = 18 + 2i

Properties Followed While the Addition of Complex Numbers

1. Closure Property

The addition of any two or more complex numbers will always yield a complex number. 

When two complex numbers are added, the resultant number is also complex. Consider two complex numbers x and y, here the value of x = a + bi, and y = c + di.

The sum of x and y can be stated as:

x + y = (a + bi) + (c + di) (equation 1)

Suppose the value of a=2, b=3, c= -6, and d= 5,

Substituting the value of a,b,c, and d in equation 1

x + y  = ( 2 + 3i) + ( -6 + 5i).

x + y =  2 + 3i – 6 + 5i

x + y = – 4 + 8i

2. Commutative Property

x + y = y + x 

In commutative addition, complex numbers can be flipped. For example:

Let Z1 = 2+3i

Z2 = 4 – 2i

Then Z1+ Z2 = (2+3i) + (4- 2i)

Z1+Z2= 6+i

Also,

Z2+ Z1= (4-2i) + (2+3i)

Z2 + Z1 = 6 + i

So, Z1+Z2  = Z2 + Z1

Hence, they commutate.

3. Associative Property

Consider three complex numbers a, b,and c, then:

(a +  b) + c = a + (b + c)

The order in which the items are added can be altered.

numbers,

Consider three complex numbers 

Z1 = m + ip

 Z2 = n + iq

 Z3 = l + it

(Z1+Z2)+Z3= [ (m+ip)+(n+iq)] + l+ir

(Z1+Z2)+Z3=m+ip+n+iq+l+ir

(Z1+Z2)+Z3=m+n+l+i(p+q+r)

Similarly,

Z1+(Z2+Z3)= (m+ip)+[(n+iq)+(l+ir)]

Z1+(Z2+Z3)=m+ip+n+iq+l+ir

Z1+(Z2+Z3)=m+n+l+i(p+q+r)

So, we can say that

(Z1+Z2)+Z3=Z1+(Z2+Z3)

4. Additive Identity Property

Let’s consider a complex number , 0 + i0 , called a zero complex number. When a zero complex number is added to any complex number, the sum is the complex number itself.

For example, z + 0 = z

5. Additive Inverse Property

z + ( -z) = 0

There exists an inverse for any complex number z.

Conclusion

By replacing the given function with its associated polynomial, and later combining using the standard principles of addition, the sum of functions can be performed. While performing the addition, certain principles stated above should be kept in mind.