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If $ f\left( x \right)=12{{x}^{5}} $ and $ g\left( x \right)=-3{{x}^{2}} $ , determine the value of $ \left( f.g \right)\left( -1 \right) $ if it is given $ \left( f.g \right)\left( x \right)=f\left( x \right).g\left( x \right) $ ?
(a) 36
(b) 42
(c) 32
(d) –36

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Last updated date: 20th Jun 2024
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Answer
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Hint: We start solving the problem by checking the domains of both the given functions f and g so that $ x=-1 $ is present in domains of both functions f and g. We then find the function $ \left( f.g \right)\left( x \right) $ by making use of the fact that $ \left( f.g \right)\left( x \right)=f\left( x \right).g\left( x \right) $ . We then substitute –1 in the obtained function $ \left( f.g \right)\left( x \right) $. We then make the necessary calculations to get the required value of $ \left( f.g \right)\left( -1 \right) $.

Complete step by step answer:
According to the problem, we are given that the function f and g are defined as $ f\left( x \right)=12{{x}^{5}} $ and $ g\left( x \right)=-3{{x}^{2}} $ . We need to find the value of $ \left( f.g \right)\left( -1 \right) $ .
We can see that the functions f and g are polynomials which means that the domain of both the functions are R. This means that there is no problem to find the function $ \left( f.g \right)\left( x \right) $.
Now, we know that $ \left( f.g \right)\left( x \right)=f\left( x \right).g\left( x \right) $ .
 $ \Rightarrow \left( f.g \right)\left( x \right)=\left( 12{{x}^{5}} \right).\left( -3{{x}^{2}} \right) $.
 $ \Rightarrow \left( f.g \right)\left( x \right)=-36{{x}^{7}} $ ---(1).
Now, let us substitute 1 in place of x in equation (1) to find the value of $ \left( f.g \right)\left( -1 \right) $ .
 $ \Rightarrow \left( f.g \right)\left( -1 \right)=-36{{\left( -1 \right)}^{7}} $ .
 $ \Rightarrow \left( f.g \right)\left( -1 \right)=-36\left( -1 \right) $ .
 $ \Rightarrow \left( f.g \right)\left( -1 \right)=36 $ .
So, we have found the value of $ \left( f.g \right)\left( -1 \right) $ as 36.
 $ \therefore $ The correct option for the given problem is (a).

Note:
 Whenever we get this type of problems, we first check the domains of both the given functions as the values of the functions $ \dfrac{f}{g} $, $ fg $, $ \sqrt{f} $ etc can only be found in the common domain of both the given functions. We should not make calculation mistakes while solving this problem. Similarly, we can expect problems to find the value of $ \sqrt{fg}\left( -1 \right) $ .