Math Interactive
2. Functions
2.3. Composite functions
2.3.4. Range
Important

The composite function fgfg involves first applying gg followed by ff. In determining the range of fgfg, we then take the range of gg “as if it is the domain of ff”.

To find Rfg,R_{fg}, we first find Rg.R_g. RfgR_{fg} is found by determining the range of the restricted function of ff with domain taken as RgR_g.

Visualization

g{g}
f{f}
fg{fg}
Dg{\boxed{D_{g}}}
Rg{\boxed{R_{g}}}
Rfg{\boxed{R_{fg}}}

Examples

0
x{x}
y{y}
y=g(x){y=g(x)}
(2,−7){\left(2 , -7 \right)}
0
x{x}
y{y}
y=f(x){y=f(x)}
y=−9{y=-9}
f:x↦e−x−9for x∈R,{f:x \mapsto \mathrm{e}^{-x} - 9} \allowbreak \quad \allowbreak {\textrm{for } x \in \mathbb{R},}
g:x↦−8x+9for x∈R,x>2.{g: x \mapsto - 8 x + 9} \allowbreak \quad \allowbreak {\textrm{for } x \in \mathbb{R}, x > 2.}
End of chapter ◼