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$\displaystyle (x+3)^{2}$

   Steps

   1. Substitute f into g
   2. $\displaystyle g(x)$$\displaystyle = x^{2}$$\displaystyle g(f(x))$$\displaystyle = (x+3)^{2}$
   3. Swapping the order gives a different expression
   4. $\displaystyle f(g(x))$$\displaystyle = x^{2}+3$
2. 02What is g(f(x)) − f(g(x))?$\displaystyle g(f(x))=4x+2,\; f(g(x))=4x+8$AnswerHide

   Answer$\displaystyle -6$

   Steps

   1. $\displaystyle (4x+2) - (4x+8)$$\displaystyle = -6$
   2. The x terms cancel; only the constants differ
   3. It is not 0 because the order changes the function
3. 03Viewing this as $y=u^{4}$, what is the inner $u$?$\displaystyle y=(2x+4)^{4}$AnswerHide

   Answer$\displaystyle 2x+4$

   Steps

   1. The bracket, which is computed first, is all of u
   2. $\displaystyle u$$\displaystyle = 2x+4$$\displaystyle y$$\displaystyle = u^{4}$
4. 04Write f(g(x)) as an expression:$\displaystyle f(x)=x+5, \quad g(x)=x^{2}$AnswerHide

   Answer$\displaystyle x^{2}+5$

   Steps

   1. Substitute g into f
   2. $\displaystyle f(x)$$\displaystyle = x+5$$\displaystyle f(g(x))$$\displaystyle = x^{2}+5$
   3. Swapping the order gives a different expression
   4. $\displaystyle g(f(x))$$\displaystyle = (x+5)^{2}$
5. 05What is f(g(3))?$\displaystyle f(x)=x+2, \quad g(x)=x^{2}$AnswerHide

   Answer$\displaystyle 11$

   Steps

   1. g goes first: compute g(3)
   2. $\displaystyle g(3)$$\displaystyle = 3^{2} = 9$
   3. Substitute that answer into f
   4. $\displaystyle f(9)$$\displaystyle = 9+2 = 11$
   5. Swapping the order gives a different value
   6. $\displaystyle g(f(3))$$\displaystyle = (5)^{2} = 25$
6. 06Write g(f(x)) as an expression:$\displaystyle f(x)=x+4, \quad g(x)=x^{2}$AnswerHide

   Answer$\displaystyle (x+4)^{2}$

   Steps

   1. Substitute f into g
   2. $\displaystyle g(x)$$\displaystyle = x^{2}$$\displaystyle g(f(x))$$\displaystyle = (x+4)^{2}$
   3. Swapping the order gives a different expression
   4. $\displaystyle f(g(x))$$\displaystyle = x^{2}+4$

FAQ