Function: Definition, types of functions, and applications!

Understanding the basic concept of functions is essential to our daily lives. Whether it is calculating distance, measuring an object, or predicting the stock market, we use functions in multiple ways.

So, what is a function? A function is simply defined as a set of rules that associates input values with output values. In other words, a function is a relationship between two sets that assigns a unique output value to each input value.

Read on to explore the main types of functions, their key properties, and their practical applications. You might also enjoy reading: 17 Maths Websites for High School Students to Get Ahead.

Table of Contents

What Is a Function?

A function is a relation between two sets where each element of the first set (called the domain) is related to only one element of the second set (called the range).

A function can be in various forms, such as a formula, a graph, or a table, and often, variables are represented by x and y. For example, a simple linear function can be represented as y = mx + b, where m and b are constants. This formula helps to describe the relationship between x and y.

Moreover, a function helps us solve various practical problems and is used in multiple fields, such as economics, engineering, and science. I encourage you to check out Khan Academy if you want to learn more about functions.

How Functions Work

In mathematics, functions are represented using symbols. The most common symbol used to represent a function is “f(x),” where “f” represents the function, and “x” represents the input value.

Let’s take the example of a simple function, f(x) = x + 2. Where “x” represents the input value, and when we substitute a particular value in place of “x,” such as f(3), the output value is calculated as f(3) = 3 + 2 = 5.

Essential Properties of a Function

One of the essential properties of a function is that for every input, there is only one output. In other words, a function cannot have two different outputs for the same input value.

This property is known as the “vertical line test” since it helps us determine whether a graph is a function. I encourage you to check this video to learn more about the vertical line test.

As a rule, if a vertical line crosses a graph in more than one place, then it is not a function. In the above example, if we substitute “x” as 3, the result should be the same every time, which is known as the vertical line test and is a significant feature of a function.

Let’s take a Real-Life Example to Understand Functions Better.

Suppose you’re at a café, and you want to order a drink. You tell the server what drink you want, and the server puts in your order. The machine makes your drink, and the server brings it to your table.

In this example, the café’s drink machine is the function. You place an order (domain), and the machine makes your drink (range).

Notice how the machine doesn’t make different drinks for the same order because the input (order) is related to only one output (drink)

Types of Functions

Functions can be classified based on their properties and formulas. Some of the essential types of functions are:

Linear Function: A function that forms a straight line, f(x)=a+bx, where b represents the slope of the function and a is the vertical intercept.
- Example: f(x) = 2x + 3.
Quadratic Function: A function that forms a U-shaped curve, f(x)=ax²+bx+c (where a≠0)
- Example f(x) = x² 2x+ 4.
Polynomial function of degree n: f(x)=a₀+a₁x+a₂x_²+⋯+a_nxⁿ where a₁,a₂⋯, a_n are coefficients. Check out Newcastle University for more worked-out polynomial functions.
- 4x⁶ – 3x² + 7 is an example of a polynomial function of degree 6, as 6 is the highest power of x.
  - A polynomial function of degree 1 is called a linear function.
  - A polynomial of unction degree 2 is called a quadratic function.
  - A polynomial function of degree 3 is called a cubic function.
  - A polynomial function of degree 4 is called a quartic function.
Exponential Function: A function that increases or decreases exponentially, such as f(x) = a^x.
- Example: f(x) = 2xe^x
Logarithmic function: A logarithmic function is defined as an inverse function to exponentiation.
- For x > 0 , a > 0, and a ≠1, y= log_a x if and only if x = a^y
Rational function: f(x)=g(x)/h(x), where g(x)≠0 and h(x)≠0.
- For example: (3x + 1)/(2x² +5)

This is just the tip of the iceberg when it comes to the types of functions. There are trigonometric functions, piecewise functions, and many more.

Practical Applications of Functions

Functions play a crucial role in various fields, including economics, engineering, and science. For instance, the stock market uses functions to predict stock prices.

The formula for calculating compound interest is a function.
And the movement of a pendulum can be expressed as a function.

Moreover, functions can help us solve practical everyday problems. For example, calculating the total cost of gas for a road trip is a function that depends on the distance traveled and the price of gas.

Furthermore, functions can be used to solve many different types of problems. For instance, a function can help us predict future values based on a series of past values.

Also, a function can be used to find maximum or minimum values or determine where two graphs intersect. In calculus, functions are used to calculate integrals and derivatives, which are essential in physics, engineering, and other sciences.

Representing a Function

Functions can be represented in different ways, including graphs, tables, and equations.

Graphs are a visual representation of a function that shows how the domain values are related to the range values.
Tables list the domain and range values of a function in an organized manner.
Equations are algebraic expressions that represent the relationship between the domain and range.

Functions can be both linear (where the rate of change is constant) or nonlinear (where the rate of change varies).

Why Function Is Important?

Functions are used in many fields of study and real-life applications. In math, functions help us determine the relationship between two variables. For example, a car’s speed is a function of the distance traveled.

In physics, functions describe how one variable affects another, like how time affects velocity. In economics, functions explain how different variables, like prices and quantities, are related.

Composition Function

In mathematics, we can also combine functions to create more complex functions, known as composition (composite function), and it can help us solve more complicated problems.

For example, if we have two linear functions, f(x) = 2x + 3 and g(x) = 3x – 1, we can find the composition of these functions by substituting one function into the other. This composition takes the form of f(g(x)) = 2(3x – 1) + 3 = 6x + 1. Check out this video to learn more about composite Functions.

Final Thoughts

A function is a machine-like relation between two sets where each input (domain) is related to only one output (range). Functions are represented in many ways, including graphs, tables, and equations, and can be linear or nonlinear.

Moreover, functions are essential in various fields of study, such as math, physics, and economics.