Yes, you can compose functions with different input and output types as long as the output type of the first function matches the input type of the second. For example, if function f maps an integer to a string (f: int → string), and function g maps a string to a list of characters (g: string → list), then their composition g ○ f is valid and has the type int → list. This is common in real-world programming, where data often flows through a pipeline of functions, each transforming the data in a different way. In practice, especially in statically typed functional languages like Haskell, the type checker enforces these constraints automatically. In dynamically typed languages like Python, it becomes the programmer’s responsibility to ensure the types match. If the types don’t align properly, composition will lead to runtime errors or unexpected behaviour, making understanding the function signatures critical to correct composition.

Function composition naturally supports immutability and side-effect-free programming because each composed function processes data without modifying it in place. Instead of changing values, each function takes an input, produces a new output, and passes it along the chain. This approach avoids shared state and makes it easier to reason about how data flows through a program. Since functions in a composition don’t perform actions like printing to the screen or changing variables elsewhere in the program, they avoid side effects, which are changes observable outside the function. This predictability is a key feature of functional programming. It means the same input will always produce the same output, regardless of when or how often the function is called. Function composition encourages the writing of small, pure functions that can be combined flexibly, increasing the modularity and reliability of code, especially in large-scale or concurrent systems where mutable state and side effects can lead to bugs.

While both function composition and method chaining involve passing the output of one operation as the input to the next, they differ significantly in structure and paradigm. Function composition comes from functional programming and is about combining standalone functions where the result of one function becomes the input to another. It focuses on pure data transformations and mathematical reasoning. On the other hand, method chaining is an object-oriented technique where each method returns the object itself or another object, allowing the next method to be called on it. For example, in Python you might write s.strip().lower().replace(" ", ""), chaining methods of a string object. In function composition, the focus is on combining named functions using operators or higher-order functions, such as compose(f, g). In contrast, method chaining depends on object state and encapsulation. Composition supports purity and reusability, while method chaining is more syntactic and tied to object interfaces.

In Haskell, function composition is implemented using the dot operator (.). This operator is actually a higher-order function itself. It takes two functions as arguments and returns a new function that applies them in order. The definition is: (.) :: (b → c) → (a → b) → a → c, which means it takes a function from b to c and another from a to b, returning a function from a to c. Under the hood, Haskell uses currying and lazy evaluation, so function composition doesn’t perform any computation until the final function is called with an input. The dot operator works by nesting function calls, effectively rewriting g ○ f as \x → g(f(x)). This efficient and compact representation avoids unnecessary memory use and supports function reuse. The compiler can also optimise chains of composed functions, making them more performant. This makes function composition a first-class citizen in Haskell, tightly integrated into the language’s core design.

Yes, while function composition works well with pure functions, it becomes more complex when dealing with functions that involve side effects such as input/output operations, network calls, or state changes. These side-effect-heavy functions don’t guarantee consistent output for a given input, which breaks the principle of referential transparency central to functional composition. In real-world programs, especially those involving user interaction, randomness, or time-based logic, composing functions must be managed carefully. In purely functional languages like Haskell, this issue is handled using monads, which encapsulate side effects and allow controlled composition through bind operators. In imperative languages, developers may separate pure logic from side-effectful operations to retain compositional benefits. Alternatively, decorators or wrapper functions may be used to convert impure functions into composable units. Ultimately, while composition is powerful, it’s not a universal solution and requires thoughtful design when integrating with systems that do not adhere strictly to functional purity.