Question

Let $f\left( x \right)=3x-7$ and $g\left( x \right)=-2x-6$, then how do you find the value of $\left( f\circ g \right)\left( 4 \right)$?

Answer 1

Hint: We start solving the problem by using the fact that $\left( f\circ g \right)\left( x \right)=f\left( g\left( x \right) \right)$ to proceed finding the composite function $\left( f\circ g \right)\left( x \right)$. We then make the necessary calculations to find the function $\left( f\circ g \right)\left( x \right)$. We then substitute $x=4$ in the obtained composite function $\left( f\circ g \right)\left( x \right)$ to proceed through the problem. We then make the necessary calculations to get the required value of $\left( f\circ g \right)\left( 4 \right)$.

Complete step by step answer:
According to the problem, we are given $f\left( x \right)=3x-7$ and $g\left( x \right)=-2x-6$. We need to find the value of $\left( f\circ g \right)\left( 4 \right)$.
Let us first find the composite function $\left( f\circ g \right)\left( x \right)$.
We know that $\left( f\circ g \right)\left( x \right)=f\left( g\left( x \right) \right)$.
So, we have $\left( f\circ g \right)\left( x \right)=3\left( g\left( x \right) \right)-7$ ---(1).
Let us substitute $g\left( x \right)=-2x-6$ in equation (1).
$\Rightarrow \left( f\circ g \right)\left( x \right)=3\left( -2x-6 \right)-7$.
$\Rightarrow \left( f\circ g \right)\left( x \right)=-6x-18-7$.
$\Rightarrow \left( f\circ g \right)\left( x \right)=-6x-25$ ---(2).
Now, let us substitute $x=4$ in equation (2) to find the value of $\left( f\circ g \right)\left( 4 \right)$.
$\Rightarrow \left( f\circ g \right)\left( 4 \right)=-6\left( 4 \right)-25$.
$\Rightarrow \left( f\circ g \right)\left( 4 \right)=-24-25$.
$\Rightarrow \left( f\circ g \right)\left( 4 \right)=-49$.
So, we have found the value of $\left( f\circ g \right)\left( 4 \right)$ as –49.
$\therefore $ The required value of $\left( f\circ g \right)\left( 4 \right)$ is –49.

Note:
We should not confuse $\left( f\circ g \right)\left( x \right)$ with $g\left( f\left( x \right) \right)$ instead of $f\left( g\left( x \right) \right)$, which is the common mistake done by students. We should not make calculation mistakes while solving for the composite function $\left( f\circ g \right)\left( x \right)$ as it makes us get the wrong value of $\left( f\circ g \right)\left( 4 \right)$. We can also solve the given problems as shown below:
We know that $\left( f\circ g \right)\left( x \right)=f\left( g\left( x \right) \right)$.
So, we have $\left( f\circ g \right)\left( 4 \right)=f\left( g\left( 4 \right) \right)$ ---(3).
Now, let us find the value of $g\left( 4 \right)$.
So, we have $g\left( 4 \right)=-2\left( 4 \right)-6$.
$\Rightarrow g\left( 4 \right)=-8-6$.
$\Rightarrow g\left( 4 \right)=-14$ ---(4).
Let us substitute equation (4) in equation (3).
$\Rightarrow \left( f\circ g \right)\left( 4 \right)=f\left( -14 \right)$.
$\Rightarrow \left( f\circ g \right)\left( 4 \right)=3\left( -14 \right)-7$.
$\Rightarrow \left( f\circ g \right)\left( 4 \right)=-42-7$.
$\Rightarrow \left( f\circ g \right)\left( 4 \right)=-49$.
So, the value of $\left( f\circ g \right)\left( 4 \right)$ is –49.