Question

Let \(f\) and \(g\) be two one-argument functions. The composition \(f\) after \(g\) is defined to be the function \(x \mapsto f(g(x))\). Define a procedure compose that implements composition. For example, if inc is a procedure that adds 1 to its argument, ((compose square inc) 6) 49

Short answer

The result is 49.

Step-by-step solution

Understand Function Composition

The composition of two functions, denoted as \((f \circ g)(x)\), is a process where function \(g\) is applied to the input \(x\), and function \(f\) is applied to the result of \(g(x)\). Mathematically, this is expressed as \(f(g(x))\). In this instance, \(f(x) = x^2\) and \(g(x) = x + 1\), so the composed function \((f \circ g)(x) = f(g(x)) = (g(x))^2 = (x+1)^2\).

Implement Composition Function

To create a function that performs composition, take two functions, \(f\) and \(g\), and return a new function. In pseudocode, this new function returns \(f(g(x))\) when called with an argument \(x\). Thus, the pseudocode could resemble: `compose(f, g)` that outputs `h(x)`, where \(h(x) = f(g(x))\).

Define Individual Functions

Define the increment function \(\text{inc}(x) = x + 1\) and the square function \(\text{square}(x) = x^2\). These will serve as \(g\) and \(f\) respectively in the composition \((f \circ g)(x) = f(g(x))\).

Compose Square and Inc

Using the `compose` function from Step 2, plug in the `square` and `inc` functions. This returns a new function that, when given an argument \(x\), will compute \((x+1)^2\).

Evaluate Composite Function with Input

Pass the value \(6\) to the composed function. Evaluate \((6 + 1)^2\) as follows: First compute the increment \(6+1=7\), then square the result: \(7^2 = 49\). The result of \((\text{compose square inc})(6)\) is therefore \(49\).

Key concepts

Programming Concepts

In programming, we often need to perform complex operations by combining simpler functions. This is especially true when utilizing function composition. Function composition is a powerful concept where you create a new function by combining two or more functions. In essence, you pass the output of one function as the input to another.

By doing this, you can build more advanced functionality without modifying the original functions. Think of it like creating a pipeline where each stage represents a function. This approach boosts modularity and reusability, allowing programmers to handle complex problems more efficiently.

• Create robust programs through modular code.
• Enhance program reusability and maintainability.
• Simplify complex operations by breaking them into smaller tasks.

Mathematical Functions

Mathematical functions are at the core of many programming tasks. A function, in mathematics, is a relation that uniquely associates an input to an output. The idea of function composition also originates from mathematics. When you compose two functions, say function \( f \) and \( g \), you apply them in a sequence. This process is denoted mathematically as \((f \circ g)(x) = f(g(x))\).

If \( g(x) = x + 1 \) and \( f(x) = x^2 \), composing them results in: \[ (f \circ g)(x) = f(g(x)) = (x+1)^2 \]

Through this, you combine operations into a single expression, streamlining calculations. Function composition also serves as a helpful abstraction tool in computer science and occurs frequently in areas such as calculus and algebra.

• Combine multiple operations into a single expression.
• Simplify calculations with transformative steps.
• Act as a foundational concept in calculus and algebra.

Procedure Definition

In programming, defining a procedure allows us to encapsulate logic into a function, making our code more organized and maintainable. A procedure definition usually involves naming the procedure, specifying parameters, and detailing the operations it performs.

To define a procedure for function composition, you need to combine two functions \( f \) and \( g \). You can create a new function \( h \), such that \( h(x) = f(g(x)) \). This promotes the use of higher-order functions, which are an advanced feature of many programming languages, including those discussed in the "Structure and Interpretation of Computer Programs" (SICP).

• Increase code clarity and organization.
• Enable code reusability with defined logic components.
• Illustrate the use of higher-order functions effectively.

SICP

"Structure and Interpretation of Computer Programs" (SICP) is a classic textbook that explores programming principles through the lens of computation. One of its focal points is the abstraction of procedures, such as function composition, to demonstrate elegant ways of solving problems.

SICP encourages thinking about programs not just as a set of instructions but as expressions and procedures that form a robust framework for problem-solving. This includes creating functions that operate on other functions, as seen with the "compose" procedure.

The text demonstrates how composing procedures can simplify complex tasks by breaking them down into more manageable parts, showing why it's influential in computer science education.

• Promote deeper understanding of programming concepts.
• Emphasize the use of abstraction in code design.
• Encourage building systems with scalable, composable components.