91 lines
2.9 KiB
Markdown
91 lines
2.9 KiB
Markdown
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# Functions
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As you might've guessed, the foundations of functional programming are
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_functions_! Yes, exactly in the math sense. If high school math is getting a
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bit hazy for you, here's a recap:
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- Functions take a single input (or multiple inputs, but we'll get into this a
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bit later!) and produce a single output
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- f(1) = 2
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- f(2) = 4
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- f(3) = 6
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- Multiple inputs are allowed to produce a single output
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- g(1) = 1
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- g(2) = 1
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- g(3) = 2
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- A single output cannot produce two different values!
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- h(1) = 1
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- h(2) = 2
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This isn't a function. Think of functions as being this unchangeable binding
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between one input and a certain output. If you evaluate the function again, it
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should always produce that same output, so it doesn't make sense for a
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function to simultaneously give two outputs for a single input. This property
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is called function _purity_.
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> There are also impure functions, which have _side effects_. For now, we will
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> focus on pure functions.
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> Example questions:
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>
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> - Which of the following are functions? (give a few functions)
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## Functions can call other functions
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The output of a function is really just like any other expression that you could
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write. For example:
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- 1, 5+5, log(x) is a valid output. f(x) = 1, g(x) = 5+5, h(x) = log(x)
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- Any expression involving the input is valid. f(x) = 2 \* x
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- Calling the function again is valid! f(x) = f(x - 1)
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- _There are a few restrictions to this. We'll see how this works later_
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In general, we can say that function calls are part of what we consider
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expressions. Since the right side of functions are also expressions, we can call
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functions inside functions!
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- See also: Expressions
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## Functions are first class
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In traditional arithmetic, we think of functions as rather _privileged_.
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Functions are these operations that we can _do_ to numbers, which are different
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from numbers.
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Well, not anymore. Functions themselves are also expressions now! We can talk
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about passing them around to other functions, but not much in the realm of
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arithmetic.
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- Venn diagram:
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- Functions
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- Can be called
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- Both
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- Can be passed around as variables
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- Numbers
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- Can be added / subtracted / multiplied / divided
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- Can take square roots and compute lots of functions
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If you noticed, all we really added was this middle part. But this gives us a
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lot of power to use functions. Say you wanted to write a function called
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**repeat**. All it does is repeat a function several times. For example,
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- If I have f(x) = 2x, then repeat(f, 5) = f(f(f(f(f(x))))) = 32x
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- If I have g(x) = 3x - 1, then repeat(g, 2) = 3(3x - 1) - 1 = 9x - 4
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In this case, the function **repeat** is known as a _higher order function_. It
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takes a function as its parameter, and can freely call it or pass it around.
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There are a few very useful and popular higher order functions:
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- map
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- filter
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- fold
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## Recursion
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## Currying
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