Range of composite functions

One method to find the range of composite functions is to first find its definition, and then find its range, being mindful of its domain.

However, this method is often potentially tricky to execute. In particular, the graph of $y = f g (x)$ is often more complicated than the graphs of the individual functions $f$ and $g .$

We will thus consider the following method to get the range of the composite function.

Finding range of composite functions

To find the range of the composite function $f g,$

Find $R_{g},$ the range of the “first” (inner) function $g$
Use $R_{g}$ as the domain of the “second” (outer) function $f$
The range of this restricted function is $R_{f g},$ the range of the composite function $f g$

Example

Question:
The functions $f$ and $g$ are given by

$\begin{aligned} f : x \mapsto x^{2}, & x \in ℝ, \\ g : x \mapsto 3 - x, & x \in ℝ, 1 < x \leq 4 . \end{aligned}$

Find the range of the composite function $f g .$

Solution:

gx-graph

From the graph above, the range of $g$ is given by $R_{g} = [- 1, 2) .$

We then restrict the domain of $f$ to $[- 1,3) .$
In the graph below, we represent this by the solid line, as opposed to the dashed lines representing the rest of $f$ following its original domain.

fx-graph

Considering the range of this restricted function, $R_{f g} = [0, 4) . ■$

Remark 1: Observe how $R_{f g}$ is different from $R_{f} = [0, \infty) .$

Remark 2: Upon finding $R_{g} = (- 1,2],$ It is tempting to skip the graph of $y = g (x)$ and only sub the end-points into $f (x)$ to get $f (- 1) = 1$ and $f (2) = 4 .$ Observe how this does not get us the final range $R_{f g} = [0, 4)$ due to the minimum point on the graph.