Question

Use the table to evaluate each expression.

(a) \(f\left( {g\left( 2 \right)} \right)\) (b) \(g\left( {f\left( 0 \right)} \right)\) (c) \(f \circ g\left( 0 \right)\) (d) \(g \circ f\left( 6 \right)\) (e) \(g \circ g\left( { - 2} \right)\) (f) \(f \circ f\left( 4 \right)\)

Short answer

(a) The value of \(f\left( {g\left( 2 \right)} \right)\) from functional values in the given table is 4.

(b) The value of \(g\left( {f\left( 0 \right)} \right)\) from functional values in the given table is 3.

(c) The value of \(f \circ g\left( 0 \right)\) from functional values in the given table is 0.

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

(e) The value of \(g \circ g\left( { - 2} \right)\) from functional values in the given table is 4.

(f) The value of \(f \circ f\left( 4 \right)\) from functional values in the given table is -2.

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 given in the following figure.

Composition of two functions

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

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

The value of \(f\left( {g\left( 2 \right)} \right)\)

(a).

From the figure,

\(\begin{aligned}f\left( {g\left( 2 \right)} \right) &= f\left( 5 \right)\\ &= 4\end{aligned}\)

Hence, the value of \(f\left( {g\left( 2 \right)} \right)\) is 4.

The value of \(g\left( {f\left( 0 \right)} \right)\)

(b).

\(\begin{aligned}g\left( {f\left

From the figure, ( 0 \right)} \right) &= g\left( 0 \right)\\ &= 3\end{aligned}\)

Hence, the value of \(g\left( {f\left( 0 \right)} \right)\) is 3.

The value of \(f \circ g\left( 0 \right)\)

(c).

From the figure and equation (1),

\(\begin{aligned}f \circ g\left( 0 \right) &= f\left( {g\left( 0 \right)} \right)\\ &= f\left( 3 \right)\\ &= 0\end{aligned}\)

Hence, the value of \(f \circ g\left( 0 \right)\) is 0.

The value of \(g \circ f\left( 6 \right)\)

(d).

From the figure and equation (1),

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

But the function \(g\left( x \right)\) is undefined at\(x = 6\).

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

The value of \(g \circ g\left( { - 2} \right)\)

(e).

From the figure and equation (1),

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

Hence, the value of \(g \circ g\left( { - 2} \right)\) is 4.

The value of \(f \circ f\left( 4 \right)\)

(f).

From the figure and equation (1),

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

Hence, the value of \(f \circ f\left( 4 \right)\) is -2.