audio read-through Writing a Function as a Composition

When you're given a multi-step task to perform, you may want to break it into pieces, and assign different pieces to different people.

When you're given a function that does several things, you may want to break it into ‘smaller’ functions that accomplish the same job!

Example (Breaking a Composite Function Into Pieces)

Consider the function: $$h(x) = 5(x-4)^3 - 7$$

This function $\,h\,$ does the following:

  1. Subtracts $\,4\,$
  2. Cubes the result
  3. Multiplies by $\,5\,$
  4. Subtracts $\,7\,$

Break these four tasks into two pieces, as shown in the mapping diagram below:

breaking a composite function into pieces

With these assignments, the composite function $\,g\circ f\ $ (where $\,f\,$ acts first, followed by $\,g\,$) accomplishes the same thing as $\,h\,$:

$$ \begin{align} &\cssId{s24}{(g\circ f\,)(x)}\cr\cr &\qquad \cssId{s25}{= g(f(x))}\cr\cr &\qquad \cssId{s26}{= g\bigl((x-4)^3\bigr)}\cr\cr &\qquad \cssId{s27}{= 5(x-4)^3 - 7}\cr\cr &\qquad \cssId{s28}{= h(x)} \end{align} $$

You Can Break a Task Into Pieces In Different Ways!

Of course, you can delegate the responsibilities in different ways:

In both cases, be sure to check that $\,(g\circ f\,)(x) = h(x)\,.$

You Can Use More Than Two Helpers!

Or, you can use more ‘helper’ functions. For example:


$$ \begin{align} &\cssId{s47}{(c\circ b\circ a\,)(x)}\cr\cr &\qquad \cssId{s48}{= c(b(a(x)))}\cr\cr &\qquad \cssId{s49}{= c(b(x-4))}\cr\cr &\qquad \cssId{s50}{= c((x-4)^3)}\cr\cr &\qquad \cssId{s51}{= 5(x-4)^3 - 7}\cr\cr &\qquad \cssId{s52}{= h(x)} \end{align} $$

The exercises in this lesson give you practice with this process of writing a function as a composition.

Concept Practice