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How do you find the value of \[2f\left( 1 \right) + 3g\left( 4 \right)\] if \[f\left( x \right) = 3x\] and $g(x) = - 4{x^2}$?

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Last updated date: 26th Feb 2024
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Hint: Put $x = 1$ in \[f\left( x \right)\] to find the value of \[f\left( 1 \right)\] and put $x = 4$ in \[g\left( x \right)\] to find the value of \[g\left( 4 \right)\]. Then put the values of \[f\left( 1 \right)\] and \[g\left( 4 \right)\] in the expression \[2f\left( 1 \right) + 3g\left( 4 \right)\] to calculate its numerical value.

Complete step by step answer:
According to the question, we have to calculate the value of the expression \[2f\left( 1 \right) + 3g\left( 4 \right)\] and two functions are given to us.
The two functions are \[f\left( x \right) = 3x\] and $g(x) = - 4{x^2}$.
First we will calculate the value of \[f\left( 1 \right)\]. This can be determined by substituting $x = 1$ in \[f\left( x \right)\]. Doing so, this will give us:
 $
   \Rightarrow f\left( 1 \right) = 3\left( 1 \right) \\
   \Rightarrow f\left( 1 \right) = 3{\text{ }}.....{\text{(1)}} \\
 $
Next we will calculate the value of \[g\left( 4 \right)\]. In the similar way, this can be obtained by substituting $x = 4$ in \[g\left( 4 \right)\]. So this will give us:
$
   \Rightarrow g\left( 4 \right) = - 4{\left( 4 \right)^2} \\
   \Rightarrow g\left( 4 \right) = - 4 \times 16 \\
   \Rightarrow g\left( 4 \right) = - 64{\text{ }}.....{\text{(2)}} \\
 $
As we know, we have to determine the value of the expression \[2f\left( 1 \right) + 3g\left( 4 \right)\]. Thus putting values of \[f\left( 1 \right)\] and \[g\left( 4 \right)\] from the equations (1) and (2) in the expression, we’ll get:
\[
   \Rightarrow 2f\left( 1 \right) + 3g\left( 4 \right) = 2 \times 3 + 3 \times \left( { - 64} \right) \\
   \Rightarrow 2f\left( 1 \right) + 3g\left( 4 \right) = 6 - 192 \\
   \Rightarrow 2f\left( 1 \right) + 3g\left( 4 \right) = - 186 \\
 \]
Thus the value of the expression is -186 and this is the answer.

Note: If we have to find the values of the function at the same value of $x$ then we can combine both the functions to solve the expression. For example if we have to determine the value of the expression \[2f\left( 1 \right) + 3g\left( 1 \right)\] or the value of the expression \[2f\left( 4 \right) + 3g\left( 4 \right)\], we can combine both the functions before putting the value of $x$ and it will become:
$
   \Rightarrow 2f\left( x \right) + 3g\left( x \right) = 2\left( {3x} \right) + 3\left( { - 4{x^2}} \right) \\
   \Rightarrow 2f\left( x \right) + 3g\left( x \right) = 6x - 12{x^2} \\
 $
Now we can easily substitute the given value of $x$.
But in expression \[2f\left( 1 \right) + 3g\left( 4 \right)\], we have to determine the values of the functions at different values of $x$. That’s why we need to solve them separately and put those values in the expression as we did above in the solution.