Example 5 from Algebra and Trigonometry, 3.4 Composition of Functions

Using a Table to Evaluate a Composite Function

Using Table 1, evaluate $f (g (3))$ and $g (f (3)) .$

$x$	$f (x)$	$g (x)$
1	6	3
2	8	5
3	3	2
4	1	7

Table 1

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
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 $f (2) = 8.$

$\begin{matrix} g (3) & = & 2 \\ f (g (3)) & = & f (2) = 8 \end{matrix}$
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 $g,$ we can evaluate

$g (f (3)) = g (3) = 2$

Table 2 shows the composite functions $f \circ g$ and $g \circ f$ as tables.

$x$ $g (x)$ $f (g (x))$ $f (x)$ $g (f (x))$

3 2 8 3 2

Table 2

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Using Table 1, evaluate $f (g (1))$ and $g (f (4)) .$

How did it go?