Question

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

Answer 1

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.