Question

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

Answer 1

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.