Question

Use the table to evaluate each expression.

(a) \(f\left( {g\left( 1 \right)} \right)\) (b) \(g\left( {f\left( 1 \right)} \right)\) (c) \(f\left( {f\left( 1 \right)} \right)\) (d) \(g\left( {g\left( 1 \right)} \right)\) (e) \(g \circ f\left( 3 \right)\) (f) \(f \circ g\left( 6 \right)\)

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

Short answer

(a) The value of\(f\left( {g\left( 1 \right)} \right)\)from functional values in the given table is 5.

(b) The value of \(g\left( {f\left( 1 \right)} \right)\) from functional values in the given table is 2.

(c) The value of \(f\left( {f\left( 1 \right)} \right)\)from functional values in the given table is 4.

(d) The value of \(g\left( {g\left( 1 \right)} \right)\) from functional values in the given table is 3.

(e) The value of \(g \circ f\left( 3 \right)\) from functional values in the given table is 1.

(e) The value of \(f \circ g\left( 6 \right)\) from functional values in the given table is 4.

Step-by-step solution

Given data

The values of the functions \(f\left( x \right)\)and \(g\left( x \right)\) for various values of \(x\) is,

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

Composition of two functions

For two functions \(f\left( x \right)\)and \(g\left( x \right)\) their composition is defined as:

\(f \circ g = f\left( {g\left( x \right)} \right)\) …… (1)

The value of

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

From the table, \(f\left( {g\left( 1 \right)} \right)\) is find as:

\(\begin{aligned}f\left( {g\left( 1 \right)} \right) &= f\left( 6 \right)\\ &= 5\end{aligned}\)

Hence, the value of \(f\left( {g\left( 1 \right)} \right)\) is 5.

The value of

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

From the table, \(g\left( {f\left( 1 \right)} \right)\) is find as:

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

Hence, the value of \(g\left( {f\left( 1 \right)} \right)\) is 2.

The value of

From the table, \(f\left( {f\left( 1 \right)} \right)\) is find as:

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

Hence, the value of \(f\left( {f\left( 1 \right)} \right)\) is 4.

The value of

\(g\left( {g\left( 1 \right)} \right)\)

From the table,\(g\left( {g\left( 1 \right)} \right)\) is find as:

\(\begin{aligned}g\left( {g\left( 1 \right)} \right) &= g\left( 6 \right)\\ &= 3\end{aligned}\)

Hence, the value of \(g\left( {g\left( 1 \right)} \right)\) is 3.

The value of

\(g \circ f\left( 3 \right)\)

From the table and equation (1), \(g \circ f\left( 3 \right)\) is find as:

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

Hence, the value of \(g \circ f\left( 3 \right)\) is 1.

The value of

\(f \circ g\left( 6 \right)\)

From the table and equation (1), \(f \circ g\left( 3 \right)\) is find as:

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

Hence, the value of \(f \circ g\left( 6 \right)\) is 4.