Meaning of a composite function f(g(x))

The composite function $f(g(x))$ applies $g$ to $x$ first, then $f$ to the result. Changing the order changes the function.

For differentiation, practice seeing $y=(2x+4)^4$ as an outer function $y=u^4$ and an inner function $u=2x+4$. The chain rule multiplies the derivatives of the outer and inner parts.

Examples

1. 01Write g(f(x)) as an expression:$\displaystyle f(x)=x+3, \quad g(x)=x^{2}$AnswerHide

   Answer$\displaystyle (x+3)^{2}$
2. 02What is g(f(x)) − f(g(x))?$\displaystyle g(f(x))=4x+2,\; f(g(x))=4x+8$AnswerHide

   Answer$\displaystyle -6$
3. 03Write g(f(x)) as an expression:$\displaystyle f(x)=x+4, \quad g(x)=x^{2}$AnswerHide

   Answer$\displaystyle (x+4)^{2}$
4. 04Viewing this as $y=u^{4}$, what is the inner $u$?$\displaystyle y=(2x+4)^{4}$AnswerHide

   Answer$\displaystyle 2x+4$
5. 05Write f(g(x)) as an expression:$\displaystyle f(x)=x+5, \quad g(x)=x^{2}$AnswerHide

   Answer$\displaystyle x^{2}+5$
6. 06What is g(f(x)) − f(g(x))?$\displaystyle g(f(x))=3x+5,\; f(g(x))=3x+15$AnswerHide

   Answer$\displaystyle -10$

FAQ

Q1Are f(g(x)) and g(f(x)) the same?Show answerHide

A

Usually not. With $f(x)=x+3$ and $g(x)=x^2$, $f(g(x))=x^2+3$ but $g(f(x))=(x+3)^2$. The order matters.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-04 · Updated 2026-09-05