Composition of Functions

Composition of functions means that we can put a function inside a function. There are two equivalent ways to write composition of functions.

    \[f(x)=2x \hspace{.25in} g(x)=x^3\]

    \[[f \circ g](x)= f[g(x)]=f(x^3)\]

3-step process

Step 1) Write out f(x) with parenthesis around where the x would be

    \[2(\hspace{.5cm})\]

Step 2) Fill in the blank spots with g(x)

    \[2(x^3)\]

Step 3) Combine like-terms and simplify

    \[2x^3\]

 

These are not commutative. Therefore it’s going to be different if we write

    \[[g \circ f](x)\]

Step 1) Write g(x) with parenthesis around where the x would be

    \[(\hspace{.5cm})^3\]

Step 2) Fill in the blank spots with f(x)

    \[(2x)^3\]

Step 3) Combine like-terms and simplify

    \[2^3x^3=8x^3\]