IB Maths AI 2.5 Notes

Composite functions

In this section, we extend our work on functions by studying composite functions and inverse functions.

Composite functions describe what happens when the output of one function is used as the input of another. Inverse functions reverse the effect of a function, so that the original input is recovered.

If the output of $g$ is used as the input of $f$, the composite function is written as

$\left(f\circ g\right)(x)=f(g(x))$

The order matters. The function $f(g(x))$ means apply $g$ first, then apply $f$.

In general, $f(g(x))$ is not the same as $g(f(x))$.

Suppose $f(x)=2x+1$ and $g(x)=x^2$, find $(f\circ g)(x)$ and $(g\circ f)(x)$.

$f(g(x))=f(x^2)=2x^2+1$.

Now reverse the order:

$g(f(x))=g(2x+1)=(2x+1)^2$.

So $(f\circ g)(x)\ne(g\circ f)(x)$.

This shows that composition depends on the order in which the functions are applied.

Composite functions are useful when one quantity depends on a second quantity, which itself depends on a third quantity.

For example, suppose the cost of producing $x$ items is $C(x)=5x+20$, and the number of items produced after $t$ hours is $N(t)=3t$.

Then the cost after $t$ hours is found by substituting $N(t)$ into $C$:

$(C\circ N)(t)=C(N(t))=C(3t)=5(3t)+20=15t+20$.

So the composite function gives the cost directly in terms of time.