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Question: Let \(f\left( x \right)=3x-7\) and \(g\left( x \right)=-2x-6\), then how do you find the value of \(...

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

Explanation

Solution

We start solving the problem by using the fact that (f∘g)(x)=f(g(x))\left( f\circ g \right)\left( x \right)=f\left( g\left( x \right) \right) to proceed finding the composite function (f∘g)(x)\left( f\circ g \right)\left( x \right). We then make the necessary calculations to find the function (f∘g)(x)\left( f\circ g \right)\left( x \right). We then substitute x=4x=4 in the obtained composite function (f∘g)(x)\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 (f∘g)(4)\left( f\circ g \right)\left( 4 \right).

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

Note:
We should not confuse (f∘g)(x)\left( f\circ g \right)\left( x \right) with g(f(x))g\left( f\left( x \right) \right) instead of f(g(x))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 (f∘g)(x)\left( f\circ g \right)\left( x \right) as it makes us get the wrong value of (f∘g)(4)\left( f\circ g \right)\left( 4 \right). We can also solve the given problems as shown below:
We know that (f∘g)(x)=f(g(x))\left( f\circ g \right)\left( x \right)=f\left( g\left( x \right) \right).
So, we have (f∘g)(4)=f(g(4))\left( f\circ g \right)\left( 4 \right)=f\left( g\left( 4 \right) \right) ---(3).
Now, let us find the value of g(4)g\left( 4 \right).
So, we have g(4)=−2(4)−6g\left( 4 \right)=-2\left( 4 \right)-6.
⇒g(4)=−8−6\Rightarrow g\left( 4 \right)=-8-6.
⇒g(4)=−14\Rightarrow g\left( 4 \right)=-14 ---(4).
Let us substitute equation (4) in equation (3).
⇒(f∘g)(4)=f(−14)\Rightarrow \left( f\circ g \right)\left( 4 \right)=f\left( -14 \right).
⇒(f∘g)(4)=3(−14)−7\Rightarrow \left( f\circ g \right)\left( 4 \right)=3\left( -14 \right)-7.
⇒(f∘g)(4)=−42−7\Rightarrow \left( f\circ g \right)\left( 4 \right)=-42-7.
⇒(f∘g)(4)=−49\Rightarrow \left( f\circ g \right)\left( 4 \right)=-49.
So, the value of (f∘g)(4)\left( f\circ g \right)\left( 4 \right) is –49.