1.4 Composition of functions  (Page 4/9)

To evaluate $f(g(3)),$ we start from the inside with the input value 3. We then evaluate the inside expression $g(3)$ using the table that defines the function $g:$ $g(3)=2.$ We can then use that result as the input to the function $f,$ so $g(3)$ is replaced by 2 and we get $f(2).$ Then, using the table that defines the function $f,$ we find that $f(2)=8.$

To evaluate $g(f(3)),$ we first evaluate the inside expression $f(3)$ using the first table: $f(3)=3.$ Then, using the table for $g\text{,\hspace{0.17em}}$ we can evaluate

[link] shows the composite functions $f\circ g$ and $g\circ f$ as tables.

To evaluate $f(g(1)),$ we start with the inside evaluation. See [link] .

We evaluate $g(1)$ using the graph of $g(x),$ finding the input of 1 on the $x\text{-}$ axis and finding the output value of the graph at that input. Here, $g(1)=3.$ We use this value as the input to the function $f.$

We can then evaluate the composite function by looking to the graph of $f(x),$ finding the input of 3 on the $x\text{-}$ axis and reading the output value of the graph at this input. Here, $f(3)=6,$ so $f(g(1))=6.$