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.

Figure-1.

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.

**Properties**

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:

**Commutativity**

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).**

**Associativity**

The **sum of functions** is also **associative**. This means that if ** f(x)**,

**, and**

`g(x)`

**are three functions, then**

`h(x)`

**.**

`((f + g) + h)(x) = (f + (g + h))(x)`

**Identity**

The **zero function**, denoted as ** 0**, where

**for all**

`0(x) = 0`

**in the**

`x`

**domain**, acts as the identity for

**function addition**. If

**is a function, then**

`f(x)`

**.**

`(f + 0)(x) = f(x)`

**Inverse**

Every function ** f(x)** has an

**additive inverse**, such that when the function and its

`-f(x)`

**inverse**are added, the result is the

**zero function**:

**.**

`(f + (-f))(x) = 0(x)`

**Distributivity**

The **sum of functions** also has distributivity property over scalar **multiplication** (real or complex numbers). If ** f(x)** and

**are functions and**

`g(x)`

**is a scalar, then**

`c`

`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)`

.

**Exercise **

**Example 1**

Sum the following two functions:

f(x) = x²

g(x) = x + 3

### Solution

The sum of these two functions is:

(f + g)(x) = f(x) + g(x) = x² + x + 3

Figure-2.

**Example 2**

Sum the following two** polynomial functions**:

f(x) = 3x³ – 2x + 1

g(x) = 4x² + 3x – 1

### Solution

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)

### Solution

The sum of these functions is:

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

Figure-3.

**Example 4**

Sum the following linear function and a quadratic function:

f(x) = 2x + 1

g(x) = x² – 3

### Solution

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)

### Solution

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)

### Solution

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)

### Solution

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

### Solution

The sum of these functions is:

(f + g)(x) = f(x) + g(x) = x³ + 5

**Applications **

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:

**Physics**

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**.

**Engineering**

**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**.

**Economics**

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.

**Statistics**

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**.

**Medicine**

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.*