Function Composition

Definition

Function composition means applying one function after another.

If:

f(x) = x + 1
g(x) = x * 2

then:

g(f(x))

means:

apply f first
then apply g to the result

Composition = output of one function becomes input to the next.

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Simple Example

Let:

f(x) = x + 1
g(x) = x * 2

For input 3:

f(3) = 4
g(4) = 8

So:

g(f(3)) = 8

Composition builds a multi-step transformation.

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Order Matters

Changing the order can change the result.

Using the same functions:

f(x) = x + 1
g(x) = x * 2

Apply f then g:

g(f(3)) = g(4) = 8

Apply g then f:

f(g(3)) = f(6) = 7

Different order, different result.

Function composition is usually order-sensitive.

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Compatibility

Composition only works when the output of the first function is a valid input for the second.

Example:

f: number -> word
g: word -> first letter

Then g(f(x)) can make sense.

But if:

g: number -> number

and f outputs a word, then g(f(x)) is not valid.

The output type of one function must fit the domain of the next.

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Composition As A New Function

A composition can be treated as one new function.

If:

h(x) = g(f(x))

then h combines both steps.

Example:

f(x) = x + 1
g(x) = x * 2
h(x) = (x + 1) * 2

A chain of functions can become a single function.

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Final Summary

• Function composition applies functions in sequence.
• The output of one function becomes input to the next.
• Order usually matters.
• Composition requires compatible domains and outputs.
• A composition can be treated as a new function.
• Composition builds larger transformations from smaller ones.