Question

Use the table to evaluate each expression

\(x\)	1	2	3	4	5	6
\(f\left( x \right)\)	3	1	5	6	2	4
\(g\left( x \right)\)	5	3	4	1	3	2

56.

(a) \(g\left( {g\left( {g\left( 2 \right)} \right)} \right)\)

(b) \(\left( {f \circ f \circ f} \right)\left( 1 \right)\)

(c) \(\left( {f \circ f \circ g} \right)\left( 1 \right)\)

(d) \(\left( {g \circ f \circ g} \right)\left( 3 \right)\)

Short answer

The values of the following expressions are:

a)\(g\left( {g\left( {g\left( 2 \right)} \right)} \right) = 1\).

b)\(\left( {f \circ f \circ f} \right)\left( 1 \right) = 2\).

c)\(\left( {f \circ f \circ g} \right)\left( 1 \right) = 1\).

d) \(\left( {g \circ f \circ g} \right)\left( 3 \right) = 2\).

Step-by-step solution

The composite function

Consider two functions \(f\) and \(g\), the composite function\(f \circ g\)(also called the composition of\(f\) and \(g\)) is defined by

\(\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right)\)

Use the table to evaluate the expression

a)

Use the table to evaluate the expression\(g\left( {g\left( {g\left( 2 \right)} \right)} \right)\)as shown below:

\(\begin{aligned}g\left( {g\left( {g\left( 2 \right)} \right)} \right) &= g\left( {g\left( 3 \right)} \right)\\ &= g\left( 4 \right)\\ &= 1\end{aligned}\)

Thus, the expression is\(g\left( {g\left( {g\left( 2 \right)} \right)} \right) = 1\).

b)

Use the table to evaluate the expression\(\left( {f \circ f \circ f} \right)\left( 1 \right)\)as shown below:

\(\begin{aligned}\left( {f \circ f \circ f} \right)\left( 1 \right) &= f\left( {f\left( {f\left( 1 \right)} \right)} \right)\\ &= f\left( {f\left( 3 \right)} \right)\\ &= f\left( 5 \right)\\ &= 2\end{aligned}\)

Thus, the expression is\(\left( {f \circ f \circ f} \right)\left( 1 \right) = 2\).

c)

Use the table to evaluate the expression\(\left( {f \circ f \circ g} \right)\left( 1 \right)\)as shown below:

\(\begin{aligned}\left( {f \circ f \circ g} \right)\left( 1 \right) &= f\left( {f\left( {g\left( 1 \right)} \right)} \right)\\ &= f\left( {f\left( 5 \right)} \right)\\ &= f\left( 2 \right)\\ &= 1\end{aligned}\)

Thus, the expression is\(\left( {f \circ f \circ g} \right)\left( 1 \right) = 1\).

d)

Use the table to evaluate the expression\(\left( {g \circ f \circ g} \right)\left( 3 \right)\)as shown below:

\(\begin{aligned}\left( {g \circ f \circ g} \right)\left( 3 \right) &= g\left( {f\left( {g\left( 3 \right)} \right)} \right)\\ &= g\left( {f\left( 4 \right)} \right)\\ &= g\left( 6 \right)\\ &= 2\end{aligned}\)

Thus, the expression is \(\left( {g \circ f \circ g} \right)\left( 3 \right) = 2\).