Sum of Functions – Definition, Applications, and Examples

Sum of Functions Definition Applications and Examples 1

This article explores the enthralling world of function addition, diving into its definition, properties, and varied applications.

Whether you’re a math enthusiast seeking to enrich your understanding or a curious reader venturing into the captivating depths of calculus, this exploration of function sums aims to unravel its complexity, shedding light on how this concept helps shape our understanding of the mathematical world and its practical implications.

Defining Sum of Functions

In mathematics, the sum of two functions is a new function that results from adding the outputs of the two functions for each corresponding input.

If we have two functions, f(x) and g(x), the sum of these functions, often denoted as (f + g)(x), is defined as:

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

This means for every input x into your new function (f + g)(x), you calculate the value of f(x) and g(x) separately and then add those two results together.

Generic representation of sum of functions


It’s important to note that the sum of functions concept extends beyond just two functions. We can define the sum of any number of functions similarly.


The sum of functions, as a basic operation in mathematics, possesses certain key properties that are rooted in the principles of function theory and algebra. Here are the primary properties:


The sum of two functions is commutative, meaning the order in which the functions are added does not change the result. If f(x) and g(x) are two functions, then (f + g)(x) = (g + f)(x).


The sum of functions is also associative. This means that if f(x), g(x), and h(x) are three functions, then ((f + g) + h)(x) = (f + (g + h))(x).


The zero function, denoted as 0, where 0(x) = 0 for all x in the domain, acts as the identity for function addition. If f(x) is a function, then (f + 0)(x) = f(x).


Every function f(x) has an additive inverse -f(x), such that when the function and its inverse are added, the result is the zero function: (f + (-f))(x) = 0(x).


The sum of functions also has distributivity property over scalar multiplication (real or complex numbers). If f(x) and g(x) are functions and c is a scalar, then c * (f + g)(x) = (c*f)(x) + (c*g)(x).

Distributivity over Function Multiplication

Function addition is also distributive over function multiplication. This means that (f * (g + h))(x) = (f*g)(x) + (f*h)(x).


Example 1

Sum the following two functions:

f(x) = x²

g(x) = x + 3


The sum of these two functions is:

(f + g)(x) = f(x) + g(x) = x² + x + 3

Graphical representation of of functions for fx equals 5 and gx equals x square


Example 2

Sum the following two polynomial functions:

f(x) = 3x³ – 2x + 1

g(x) = 4x² + 3x – 1


The sum of these functions is:

(f + g)(x) = f(x) + g(x) = 3x³ + 4x² + x

Example 3

Sum the following two trigonometric functions:

f(x) = sin(x)

g(x) = cos(x)


The sum of these functions is:

(f + g)(x) = f(x) + g(x) = sin(x) + cos(x)

Graphical representation of of functions for fx equals sinx and gx equals


Example 4

Sum the following linear function and a quadratic function:

f(x) = 2x + 1

g(x) = x² – 3


The sum of these functions is:

(f + g)(x) = f(x) + g(x) = x² + 2x – 2

Example 5

Sum the following exponential function and a logarithmic function:

f(x) = eˣ

g(x) = ln(x) (Natural logarithm)


The sum of these functions is:

(f + g)(x) = f(x) + g(x) = eˣ + ln(x)

Example 6

Sum the following two rational functions:

f(x) = 1/x

g(x) = 1/(x²)


The sum of these functions is:

(f + g)(x) = f(x) + g(x) = 1/x + 1/(x²)

Example 7

Sum the following two radical functions:

f(x) = √x

g(x) = √(2x)


The sum of these functions is:

(f + g)(x) = f(x) + g(x) = √x + √(2x)

Example 8

Suppose we have a function that is constant and a cubic function:

f(x) = 5

g(x) = x³


The sum of these functions is:

(f + g)(x) = f(x) + g(x) = x³ + 5


The sum of functions concept, which lies at the heart of many mathematical operations, finds use in various fields due to its inherent utility. Here are some notable applications:


In physics, multiple forces often act on an object simultaneously. Each of these forces can be represented as a function of position, time, or other variables, and the net force acting on the object is the sum of these individual force functions. The concept of the sum of functions also plays a crucial role in wave superposition, where multiple waves combine to form a resultant wave.


Engineers often need to analyze the combined effect of different factors. For example, in electrical engineering, the total current or voltage in a circuit can be the sum of the currents or voltages from multiple sources, represented as functions of time.


In economics, a firm’s total cost function is often the sum of different cost functions such as fixed costs, variable costs, etc. Similarly, a market demand function can be represented as the sum of individual demand functions.

Computer Science

In machine learning and artificial intelligence, an ensemble of models (like a neural network or a decision forest) can be seen as the sum of simpler functions or models. Each model makes a prediction (a function of the input), and these predictions are combined (often by summing) to make a final prediction.


In statistics, the concept of the sum of functions is employed when dealing with the probability density function of the sum of independent random variables.


In medicine, different risk factors can contribute to the overall disease risk. Each risk factor can be modeled as a function of various health metrics (like age, BMI, cholesterol level, etc.), and the overall risk can be modeled as the sum of these individual risk functions.

These examples illustrate that function addition is ubiquitous and immensely useful in diverse scientific fields. By understanding and applying this concept, one can analyze complex systems piecewise, breaking them down into simpler components that can be individually understood and combined.

All images were created with GeoGebra.