Textbook
Given $g:A\to B$ and $f:B\to C$, their composition $f\circ g:A\to C$ is defined by $(f\circ g)(x):=f(g(x))$. The function on the right, $g$, acts first, followed by $f$. The first output must lie in the next function’s domain.
For $g(x)=x+1$ and $f(x)=x^2$, the results are $(f\circ g)(2)=9$ but $(g\circ f)(2)=5$. Composition is generally not commutative: changing the order changes the rule. The expression $f(g(x))$ denotes the composed function’s value, not a product of values.
Domain compatibility matters more than the formula’s appearance. For $g:A\to B$, every actual value $g(x)$ must lie in the domain of $f$; otherwise composition must be restricted to inputs where the next step is defined. Three compatible functions can be regrouped because both arrangements give $f(g(h(x)))$. This preserves the application order and does not permit exchanging the functions themselves.
Definitions and further reading: Mathematics for Computer Science.
In plain language
Composition resembles the order “socks, then shoes”: the first action supplies the input to the second. Exchanging the actions preserves their names but changes the result and the mood of nearby observers. Nested notation runs from the inside outward, so the outer step receives an intermediate result. That result must suit the next function; two working machines do not guarantee compatible parts.
Example
The original illustrative rules are $g(x)=x+1$, $f(x)=x^2$, and input 2. The example traces $f\circ g$ first, then calculates the opposite order.
Function Composition: Feeding One Output Into Another
Inner function
Adding 1 to input 2 gives 3.
Step 1 of 3: Inner function
Read all steps
Inner function
Adding 1 to input 2 gives 3.
- State
- 2current3current
Outer function
Squaring the intermediate value 3 gives 9.
- State
- 3current9current
Opposite order
Squaring 2 and then adding 1 gives the result 5.
- State
- 2current4current5current
The calculation $f(g(2))$ has intermediate value 3, so squaring acts on three. In the reverse order, the intermediate value is 4 and one is added to it. Results 9 and 5 represent different composed functions rather than faulty arithmetic. The expanded rules are $(x+1)^2$ and $x^2+1$: parentheses show precisely which input the outer operation receives.
Related concepts: Functions.