Question

How do you find \[\left[ {f \circ g} \right]\left( 2 \right)\]and\[\left[ {g \circ f} \right]\left( 2 \right)\]given\[f\left( x \right) = 2x - 1,\]\[g\left( x \right) = - 3x?\]

Answer 1

Hint:This question involves the arithmetic operation like addition/ subtraction/ multiplication/ division. We need to know how to find the value of \[x\]from the terms\[\left[ {f \circ g} \right]\left( 2 \right)\]and\[\left[ {g \circ f} \right]\left( 2 \right)\]. We need to know the arithmetic functions with the involvement of different signs. Also, we need to know the basic formulae with the involvement of\[f\left( {g\left( x \right)} \right)\]and\[g\left( {f\left( x \right)} \right)\]. 39g

Complete step by step solution:
The given question is shown below,
\[\left[ {f \circ g} \right]\left( 2 \right) = ? \to \left( 1 \right)\]
\[\left[ {g \circ f} \right]\left( 2 \right) = ? \to \left( 2 \right)\]
\[f\left( x \right) = 2x - 1 \to \left( 3 \right)\]
\[g\left( x \right) = - 3x \to \left( 4 \right)\]
We know that,
\[\left[ {f \circ g} \right]\left( x \right) = f\left( {g\left( x \right)} \right) \to \left( 5 \right)\]
\[\left[ {g \circ f} \right]\left( x \right) = g\left( {f\left( x \right)} \right) \to \left( 6 \right)\]
By comparing the equation\[\left( 1 \right)\]and\[\left( 5 \right)\], we get\[x = 2\].
So, the equation\[\left( 5 \right)\]becomes,
\[\left( 5 \right) \to \left[ {f \circ g} \right]\left( x \right) = f\left( {g\left( x \right)} \right)\]
Put\[x = 2\], so we get
\[\left[ {f \circ g} \right]\left( 2 \right) = f\left( {g\left( 2 \right)} \right) \to \left( 7 \right)\]
So, we need to find
\[g\left( 2 \right) = ?\]
We know that
From \[\left( 4 \right) \to g\left( x \right) = - 3x\]
Put\[x = 2\], so we get
\[g\left( 2 \right) = - 3 \times 2 = - 6\]
So, the equation\[\left( 7 \right)\]becomes,
\[\left[ {f \circ g} \right]\left( 2 \right) = f\left( {g\left( 2 \right)} \right) = f\left( { - 6} \right)\]
We need to find\[f\left( { - 6} \right) = ?\]
We know that,
\[\left( 3 \right) \to f\left( x \right) = 2x - 1\]
Put \[x = - 6\], so we get
\[
f\left( { - 6} \right) = \left( {2 \times - 6} \right) - 1 \\
f\left( { - 6} \right) = - 12 - 1 \\
f\left( { - 6} \right) - 13 \\
\]
So, we get
\[\left[ {f \circ g} \right]\left( 2 \right) = f\left( {g\left( 2 \right)} \right) = f\left( { - 6} \right) = - 13 \to \left( A \right)\]
Next, we need to solve
\[\left[ {g \circ f} \right]\left( x \right) = g\left( {f\left( x \right)} \right)\]
By comparing the equation\[\left( 2 \right)\]and\[\left( 6 \right)\], we get\[x = 2\]
So, the equation\[\left( 6 \right)\]becomes,
\[\left( 6 \right) \to \left[ {g \circ f} \right]\left( x \right) = g\left( {f\left( x \right)} \right)\]
Put\[x = 2\]
\[\left[ {g \circ f} \right]\left( 2 \right) = g\left( {f\left( 2 \right)} \right)\]
We need to find\[f\left( 2 \right) = ?\]
We know that,
\[f\left( x \right) = 2x - 1\]
Put\[x = 2\]
\[f\left( 2 \right) = \left( {2 \times 2} \right) - 1\]
\[
f\left( 2 \right) = 4 - 1 \\
f\left( 2 \right) = 3 \\
\]
So, we get
\[\left[ {g \circ f} \right]\left( 2 \right) = g\left( {f\left( 2 \right)} \right) = g\left( 3 \right)\]
So, we need to find\[g\left( 3 \right) = ?\]
We know that,
\[g\left( x \right) = - 3x\]
Put\[x = 3\]
\[g\left( 3 \right) = - 3 \times 3\]
\[g\left( 3 \right) = - 9\]
So, we get
\[\left[ {g \circ f} \right]\left( 2 \right) = g\left( {f\left( 2 \right)} \right) = g\left( 3 \right) = - 9\]
So, the final answer is,
\[
\left[ {f \circ g} \right]\left( 2 \right) = - 13 \\
\left[ {g \circ f} \right]\left( 2 \right) = - 9 \\
\]

Note: This question involves the operation of addition/ subtraction/ multiplication/ division. To solve this type of question we would remember the formula for\[\left[ {f \circ g} \right]\left( x \right)\]and\[\left[ {g \circ f} \right]\left( x \right)\]. Also, we need to remember the following things when multiplying different sign terms,
1) When a negative term is multiplied by a negative term, the final answer will be a positive
term.
2) When a positive term is multiplied by a positive term, the final answer will be a positive
term.
3) When a negative term is multiplied with a positive term, the final answer will be a negative term.