Function composition
Function composition means plugging the output of one function into another. If you see something like (f(g(x))), evaluate the inside function first, then use that result as the input of the outside function.
Method
- Find the inner function first.
If the expression is (f(g(x))), compute (g(x)) before anything else.
- Substitute carefully.
Replace the variable in the outside function with the entire inner expression.
- Simplify step by step.
Distribute, combine like terms, and reduce fractions if needed.
- Watch the order.
In general, (f(g(x))) is not the same as (g(f(x))).
Useful checks
- Make sure every variable in the final answer is simplified as much as possible.
- If numbers are given for (x), evaluate the inside function first and then the outside one to avoid mistakes.
- If the result can be checked by direct substitution, test it with a simple value.
Common mistake to avoid
A frequent error is composing functions in the wrong order. Always read the notation exactly and follow the nesting from the inside out.