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Chapter 2: Problem 5

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

Expert verified
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

01

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.
02

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.
03

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.
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Equation \((2-30)\) is an approximation correct only if the gravitational time- dilation effect is small. In this exercise, it is also assumed to be small. but we still allow for a nonuniform gravitational field. We stan with (2-29), based on the Doppler effect in the acceleraing frame. Consider two elevations, the lower at \(r_{1}\) and the upper at \(r_{1}+d r\). Equation \((2-29)\) becomes $$ \frac{f\left(r_{1}+d r\right)}{f\left(r_{i}\right)}=\left(1-\frac{g\left(r_{1}\right) d r}{c^{2}}\right) $$ Similarly, if we consider elevations \(r_{1}+d r\) and \(r_{1}+\) \(2 d r\). we have $$ \frac{f\left(r_{1}+2 d r\right)}{f\left(r_{1}+d r\right)}=\left(1-\frac{g\left(r_{1}+d r\right) d r}{c^{2}}\right) $$ We continue the process, incrementing \(r\) by \(d r\), until we reach \(r_{2}\) $$ \frac{f\left(r_{2}\right)}{f\left(r_{2}-d r\right)}=\left(1-\frac{g\left(r_{2}-d r\right) d r}{c^{2}}\right) $$ Now imagine multiplying the left sides of all the equations and setting the product equal to the product of all the right sides. (a) Argue that the left side of the product is simply \(f\left(r_{2}\right) / f\left(r_{1}\right)\). (b) Assuming that the term \(g d r / c^{2}\) in each individual equation is very small. so that products of such terms can be ignored, argue that the right side of the product is $$ 1-\frac{1}{c^{2}} \int g(r) d r $$ (c) Deduce \(g(r)\) from Newton's universal law of gravitation, then argue that equation \((2-31)\) follows from the result, just as \((2 \cdot 30)\) does from \((2-29)\).

Planet \(W\) is 12 ly from Earth. Anna and Bob are both 20 yr old. Anna travels to Planet \(W\) at \(0.6 c\), quickly turns around, and returns to Earth at \(0.6 c\). How old will Anna and Bob be when Anna gets back?

From a standstill, you begin jogging at \(5 \mathrm{~m} / \mathrm{s}\) directly toward the galaxy Centaurus A. which is on the horizon \(2 \times 10^{23} \mathrm{~m}\) away, (a) There is a clock in Centaurus A. According to you, how will readings on this clock differ before and after you begin jogging? (Remember: You change frames.) (b) The planet Neptune is between Earth and Centaurus A. \(4.5 \times 10^{9} \mathrm{~m}\) from Earth. How much would readings on a clock there differ? (c) What would be the time differences if you had instead begun jogging in the opposite direction? (d) What do these results tell you about the observations of a traveling twin who accelerates toward his Earth-bound twin? How do these observations depend on the distance between the twins?

Appearing in the time-dilation and length-contraction formulas, \(\gamma_{y}\) is a reasonable measure of the size of relativistic effects. Roughly speaking, at what speed would observations deviate from classical expectations by \(1 \% ?\)

In the frame in which they are at rest, the number of muons at tiroe \(r\) is given by $$ N=N_{0} e^{-\nu / \tau} $$ where \(N_{0}\) is the number at \(r=0\) and \(\tau\) is the mean lifetime 2.2 \mus. (a) If muons are produced at a height of \(4.0 \mathrm{~km}\), beading toward the ground at \(0.93 \mathrm{c}\). what fraction will survive to reach the ground? (b) What fraction would reach the ground if classical mechanics were valid?
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