Question

Write a JavaScript function named compose that computes the composition of two functions. That is, compose \((f, g)\) is \(f \circ g\), where \(f\) and \(g\) are functions of one parameter. Recall that \(f \circ g\) is the function defined by \((f \circ g)(x)=f(g(x))\).

Short answer

The 'compose' function in JavaScript is a higher order function that takes two input functions \(f\) and \(g\) and returns a new function that applies \(g\) to its input \(x\) and then applies \(f\) to the result. This is the concept of function composition \(f \circ g\). The JavaScript implementation of the function is: \[ function compose(f, g) {return function(x) {return f(g(x));};} \]

Step-by-step solution

Define the compose function

Firstly, a function called 'compose' is declared. It takes two parameters, \(f\) and \(g\), which are other functions. Then, the 'compose' function returns a new function.

Return the composed function

The function that 'compose' returns takes a single parameter \(x\). Then, it returns the result of first applying \(g\) to \(x\), which is \(g(x)\), and then applying \(f\) to the result, which is \(f(g(x))\). In JavaScript, this is done by using the return keyword to give back the result of the composition of \(f\) and \(g\). Note that because 'compose' is returning a function, this is a higher-order function.

Complete JavaScript implementation

The entire function in JavaScript would look like this: \[ function compose(f, g) {return function(x) {return f(g(x));};} \] This function can now be used to compute the composition of any two functions that each take one parameter.

Testing the function

The function can be tested by passing in two simple functions, say \(f(x) = 2x\) and \(g(x) = x + 3\). Using these two functions, \(f \circ g (x)\) is equal to \(2 * ((x + 3))\). Therefore, \(f \circ g (2)\) would be \(2 * ((2+3) = 2 * 5 = 10\). In JavaScript this would look like this: \[ var f = function(x) {return 2 * x;}; var g = function(x) {return x + 3;}; var h = compose(f, g); console.log(h(2)); \] When this is executed, it should print 10 to the console.

Key concepts

JavaScript functions

In programming, functions are fundamental building blocks, especially in JavaScript. They are used to encapsulate and reuse code.

A JavaScript function is defined with the function keyword, followed by a name, a list of parameters in parentheses, and a block of code enclosed in braces that specifies the actions to be performed. For example, a simple function to add two numbers would look like this:

function add(a, b) {
return a + b;
}

Functions can take zero or more parameters and can return a value using the return statement. If no return statement is used, the function returns undefined. Parameters can include other functions, making JavaScript incredibly versatile for functional programming strategies.

Higher-order functions

A higher-order function is a function that can take other functions as arguments or return a function as a result. They are a key feature in JavaScript and enable powerful techniques like function composition and currying.

Here are some characteristics of higher-order functions:

• Can create functions on the fly.
• Can store functions as variables or properties.
• Allow for abstract or generic processing of data.

For instance, the compose function in the original exercise is a higher-order function because it returns a new function as the result of combining two provided functions.

Function operations

In JavaScript, functions can be assigned to variables, passed as arguments to other functions, and even returned from other functions. This flexibility allows for operations on functions such as composition, where the result of one function is passed as the input to another.

Function composition, as shown in the exercise, is a powerful concept where two or more functions are combined to create a new function. This is particularly useful for creating pipelines of functions where the output of one is the input to the next, effectively building complex operations from simpler ones.

Understanding function operations allows developers to write more modular, readable, and maintainable code by breaking complex operations into simpler, composable parts.

Mathematical concepts in programming

Mathematical concepts are widely used in programming, and JavaScript is no exception. Concepts such as functions, compositions, and operations are foundations of mathematical logic that apply to programming as well.

For example, function composition in mathematics involves combining two functions in such a way that the output of one function becomes the input of the other. This matches the behavior of the compose function in the JavaScript exercise. Understanding the mathematical underpinning of function composition can deepen a programmer's ability to reason about code structure and flow.

Embracing these mathematical concepts allows programmers to write code that's not just functional, but elegant and efficient, mirroring the precision and clarity of a mathematical equation.