Question

\[f \circ g\left( 9 \right) = 683\] Let \[f\] and \[g\] be functions from \[R\] to \[R\] defined as \[f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{7{x^2} + x - 8,}&{x \le 1}\\\begin{array}{l}4x + 5,\\8x + 3\end{array}&\begin{array}{l}1 < x \le 7\\x > 7\end{array}\end{array}} \right.\] , \[g\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{\left| x \right|,}&{x < - 3}\\\begin{array}{l}0,\\{x^2} + 4,\end{array}&\begin{array}{l} - 3 \le x < 2\\x \ge 2\end{array}\end{array}} \right.\]. Then which of the following option is true?
A. \[f \circ g\left( { - 3} \right) = 8\]
B. \[f \circ g\left( 9 \right) = 683\]
C. \[g \circ f\left( 0 \right) = - 8\]
D. \[g \circ f\left( 6 \right) = 427\]

Answer 1

Hint: Here, two function \[f\] and \[g\] are given. To check which option is true, solve each option by using the composition of the functions. For the first and the second options, first calculate \[g\left( x \right)\] for the corresponding numbers. Then, substitute the value of \[g\left( x \right)\] as the input of \[f\left( x \right)\] and solve it. Similarly, for the third and fourth options, first calculate \[f\left( x \right)\] for the corresponding numbers. Then, substitute the value of \[f\left( x \right)\] as the input of \[g\left( x \right)\] and solve it. In the end, verify the answers with the given options and get the required answer.

Formula Used: Composition of functions:
\[f \circ g\left( x \right) = f\left( {g\left( x \right)} \right)\]
\[g \circ f\left( x \right) = g\left( {f\left( x \right)} \right)\]

Complete step by step solution: Given:
The functions \[f\] and \[g\] from \[R\] to \[R\] are defined as \[f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{7{x^2} + x - 8,}&{x \le 1}\\\begin{array}{l}4x + 5,\\8x + 3\end{array}&\begin{array}{l}1 < x \le 7\\x > 7\end{array}\end{array}} \right.\] and \[g\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{\left| x \right|,}&{x < - 3}\\\begin{array}{l}0,\\{x^2} + 4,\end{array}&\begin{array}{l} - 3 \le x < 2\\x \ge 2\end{array}\end{array}} \right.\].
Let’s solve each option by using the rule of the composition of functions.
Option A) \[f \circ g\left( { - 3} \right) = 8\]
Apply the rule of the composition of functions.
\[f \circ g\left( { - 3} \right) = f\left( {g\left( { - 3} \right)} \right)\]
Here, first calculate the value of \[g\left( { - 3} \right)\].
From the given function \[g\left( x \right)\], we get \[g\left( { - 3} \right) = 0\].
So,
\[f \circ g\left( { - 3} \right) = f\left( 0 \right)\]
Now from the given function \[f\left( x \right)\], we get
\[f \circ g\left( { - 3} \right) = 7{\left( 0 \right)^2} + \left( 0 \right) - 8\]
\[ \Rightarrow f \circ g\left( { - 3} \right) = 0 + 0 - 8\]
\[ \Rightarrow f \circ g\left( { - 3} \right) = - 8\]
Thus, option A is incorrect.

Option B) \[f \circ g\left( 9 \right) = 683\]
Apply the rule of the composition of functions.
\[f \circ g\left( 9 \right) = f\left( {g\left( 9 \right)} \right)\]
Here, first calculate the value of \[g\left( 9 \right)\].
From the given function \[g\left( x \right)\], we get \[g\left( 9 \right) = {\left( 9 \right)^2} + 4\].
So,
\[f \circ g\left( 9 \right) = f\left( {{{\left( 9 \right)}^2} + 4} \right)\]
\[ \Rightarrow f \circ g\left( 9 \right) = f\left( {85} \right)\]
Now from the given function \[f\left( x \right)\], we get
\[f \circ g\left( 9 \right) = 8\left( {85} \right) + 3\]
\[ \Rightarrow f \circ g\left( 9 \right) = 680 + 3\]
\[ \Rightarrow f \circ g\left( 9 \right) = 683\]
Thus, option B is correct.

Option C) \[g \circ f\left( 0 \right) = - 8\]
Apply the rule of the composition of functions.
\[g \circ f\left( 0 \right) = g\left( {f\left( 0 \right)} \right)\]
Here, first calculate the value of \[f\left( 0 \right)\].
From the given function \[f\left( x \right)\], we get \[f\left( 0 \right) = 7{\left( 0 \right)^2} + 0 - 8\].
So,
\[g \circ f\left( 0 \right) = g\left( {7{{\left( 0 \right)}^2} + 0 - 8} \right)\]
\[ \Rightarrow g \circ f\left( 0 \right) = g\left( { - 8} \right)\]
Now from the given function \[g\left( x \right)\], we get
\[g \circ f\left( 0 \right) = \left| { - 8} \right|\]
\[ \Rightarrow g \circ f\left( 0 \right) = 8\]
Thus, option C is incorrect.
Option D) \[g \circ f\left( 6 \right) = 427\]
Apply the rule of the composition of functions.
\[g \circ f\left( 6 \right) = g\left( {f\left( 6 \right)} \right)\]
Here, first calculate the value of \[f\left( 6 \right)\].
From the given function \[f\left( x \right)\], we get \[f\left( 6 \right) = 4\left( 6 \right) + 5\].
So,
\[g \circ f\left( 6 \right) = g\left( {4\left( 6 \right) + 5} \right)\]
\[ \Rightarrow g \circ f\left( 6 \right) = g\left( {29} \right)\]
Now from the given function \[g\left( x \right)\], we get
\[g \circ f\left( 6 \right) = {\left( {29} \right)^2} + 4\]
\[ \Rightarrow g \circ f\left( 6 \right) = 841 + 4\]
\[ \Rightarrow g \circ f\left( 6 \right) = 845\]
Thus, option D is incorrect.
From the above solved options, we get

Option ‘B’ is correct

Note: Students sometimes make mistake and calculate \[g \circ f\left( x \right)\] as \[f \circ g\left( x \right)\]. They forget that composite function is not a commutative function. So, \[g \circ f\left( x \right) \ne f \circ g\left( x \right)\].