Formula and domain of composite functions

Finding a composite function

To get the formula of a composite function $f g,$ we take the inner function $g$ and substitute into $f .$ For example, if $f (x) = 2 x - 1$ and $g (x) = x^{2},$ then we have

$\begin{aligned} f g (x) & = f (x^{2}) \\ = 2 x^{2} - 1 \end{aligned}$

and

$\begin{aligned} g f (x) & = g (2 x + 1) \\ = (2 x - 1)^{2} \end{aligned}$

Domain of composite function

Since the composite function $f g$ starts by applying $g,$ the domain of $f g$ will be the domain of $g .$

$D_{f g} = D_{g}$
$D_{g f} = D_{f}$

Example

Question:

$\begin{aligned} f : x \mapsto 2 x - 1, & 0 \leq x < 10 \\ g : x \mapsto x^{2} + 3, & - 2 < x < 1 . \end{aligned}$

Give a definition, in similar form, for the composite function $f g .$

Solution:

$\begin{aligned} f g (x) & = f (x^{2} + 3) \\ = 2 (x^{2} + 3) - 1 \\ = 2 x^{2} + 5 \end{aligned}$

We note that $D_{f g} = D_{g} = (- 2, 1),$ so

$f g : x \mapsto 2 x^{2} + 5, - 2 < x < 1 . ∎$