Checkpoint: Composition of Functions Solution

Determine the domain of $h [g (x)]$ for the functions $g (x) = 3 x - 2$ and $h (x) = 3 x^{2} - 5$ .

$\begin{array}{rcl} h [g (x)] & = & 3 {(3 x - 2)}^{2} - 5 D {omain}_{g} : \{x | x \in ℝ\} \\ = & 3 (9 x^{2} - 12 x + 4) - 5 \\ h [g (x)] & = & 27 x^{2} - 36 x + 7 D {omain}_{h \circ g} : \{x | x \in ℝ\} \end{array}$

Note

Q: What type of functions are g and h?

A: g is a linear function, h is a quadratic (parabolic) function.

Q: Why is the order of the composition of functions important?

A: Because the composition of functions is not commutative, and the order also helps determine the domain.