We can insert one function into another to create composite functions. It's similar to "a function within a function."
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.
