Find the Range of Composite Function

Example

Given $$ f: x \mapsto e^x, \phantom{000} x \in \mathbb{R}, 0 < x \le 1 $$ $$ g : x \mapsto x^2 - 2x - 2, \phantom{000} x \in \mathbb{R}, 1 \le x \le 4 $$

and $$ D_f = (0, 1] \text{ and } R_f = (1, e] $$ $$ D_g = [1, 4] \text{ and } R_g = [-3, 6] $$

Method 1: Find range of gf by using range of f as new domain of g

Graph $y = g(x)$ by using $R_f = (1 , e] $ as the new domain:

$$ \therefore R_{gf} = (-3, e^2 - 2e - 2] $$

Method 2: Define the composite function gf and find it's range

Defining gf \begin{align} gf(x) & = g (e^x) \\ & = (e^x)^2 - 2(e^x) - 2 \\ & = e^{2x} - 2e^x - 2 \\ \\ gf : x \mapsto & \phantom{.} e^{2x} - 2e^x - 2, \phantom{000} x \in \mathbb{R}, 0 < x \le 1 \end{align}

Graph $y = gf(x)$:

$$ \therefore R_{gf} = (-3, e^2 - 2e - 2] $$