What are Composite Functions?

Substitute Function → Evaluate Function

Creating composite functions is similar to evaluating math functions: 

• We identify the function inputs and substitute them into the other, 
• Then, we solve for the resulting expression.

As an example, say we have two polynomial functions:

• $f(x)=x+2$
• $g(x)=5x$

We want to find $(f\cdot g)(x)=f(g(x))$. All we have to do is to replace the $x$ in $f(x)$ with $g(x)$:

$f(g(x))=(5x)+2$

$f(g(x))=5x+2$

To find $(g\cdot f)(x)=g(f(x))$, we replace the $x$ in $g(x)$ with $f(x)$:

$g(f(x))=5(x+2)$

$g(f(x))=5x+10$

Composite functions can be a mix of different types.

Summary

Composite functions are functions within a function.

To create composite functions, we identify the function inputs, substitute another function to it, then solve for the resulting expression.