Question

Use the given graphs of \(f\) and \(g\) to evaluate each expression, why it is undefined.

57.

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

(b) \(g\left( {f\left( {\bf{0}} \right)} \right)\)

(c) \(\left( {f \circ g} \right)\left( {\bf{0}} \right)\)

(d) \(\left( {g \circ f} \right)\left( {\bf{6}} \right)\)

(e) \(\left( {g \circ g} \right)\left( { - {\bf{2}}} \right)\)

(f) \(\left( {f \circ f} \right)\left( {\bf{4}} \right)\)

Short answer

The values of the following expressions are:

a) The expression is \(f\left( {g\left( 2 \right)} \right) = 4\).

b) The expression is \(g\left( {f\left( 0 \right)} \right) = 3\).

c) The expression is \(\left( {f \circ g} \right)\left( 0 \right) = 0\).

d) The expression is \(\left( {g \circ f} \right)\left( 6 \right) = g\left( 6 \right)\). The value is undefined.

e) The expression is \(\left( {g \circ g} \right)\left( { - 2} \right) = 4\)

f) The expression is \(\left( {f \circ f} \right)\left( 4 \right) = - 2\).

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 as:

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

Use the graph of \(f\) and \(g\) to evaluate the expression

a)

From the given graph, \(g\left( 2 \right) = 5\) since the point \(\left( {2,5} \right)\) is on the graph of \(g\).

Evaluate the expression \(f\left( {g\left( 2 \right)} \right)\) as shown below:

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

The point \(\left( {5,4} \right)\) is on the graph of \(f\).

Thus, the expression \(f\left( {g\left( 2 \right)} \right) = 4\).

b)

Evaluate the expression \(g\left( {f\left( 0 \right)} \right)\) as shown below:

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

Thus, the expression is \(g\left( {f\left( 0 \right)} \right) = 3\).

c)

Evaluate the expression \(\left( {f \circ g} \right)\left( 0 \right)\) as shown below:

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

Thus, the expression is \(\left( {f \circ g} \right)\left( 0 \right) = 0\).

d)

Evaluate the expression \(\left( {g \circ f} \right)\left( 6 \right)\) as shown below:

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

Since there is no point on the graph of \(g\) that has 6 as a \(x - \)coordinate. So, it is undefined.

Thus, the expression \(\left( {g \circ f} \right)\left( 6 \right) = g\left( 6 \right)\).

e)

Evaluate the expression \(\left( {g \circ g} \right)\left( { - 2} \right)\) as shown below:

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

Thus, the expression is \(\left( {g \circ g} \right)\left( { - 2} \right) = 4\).

f)

Evaluate the expression \(\left( {f \circ f} \right)\left( 4 \right)\) as shown below:

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

Thus, the expression is \(\left( {f \circ f} \right)\left( 4 \right) = - 2\).