Question

If $f\left( x \right) = 4 - {x^2}$ and $g\left( x \right) = 6x$ , which expression is equivalent to $\left( {g - f} \right)\left( 3 \right)$ ?

Answer 1

In the above question, we are given two different polynomial functions of the variable $x$ . The first function is given as $f\left( x \right) = 4 - {x^2}$ while the other function is written as $g\left( x \right) = 6x$ . We have to determine the value of the expression which is equivalent to the difference of these two functions, i.e. an expression which is equivalent to $\left( {g - f} \right)\left( 3 \right)$ . In order to approach the solution, we have to use the identity of the difference of the two functions of the same variable, that is given by $\left( {f - g} \right)\left( x \right) = f\left( x \right) - g\left( x \right)$. Using this identity, we can obtain the value to $\left( {g - f} \right)\left( 3 \right)$ by simply putting $x = 3$ in the obtained equation.

Complete step by step answer:
We are given two functions of the same variable $x$ written as,
$\Rightarrow f\left( x \right) = 4 - {x^2}$
And,
$\Rightarrow g\left( x \right) = 6x$
We have to find the value of the expression which is equivalent to the difference of these two functions at $x = 3$. That means we have to find the value of the function $\left( {g - f} \right)\left( 3 \right)$. Since, as we know that the difference of the two functions of the same variable as $f\left( x \right)$ and $g\left( x \right)$ , that is given by the formula,
$\Rightarrow \left( {f - g} \right)\left( x \right) = f\left( x \right) - g\left( x \right)$
Hence,
$\Rightarrow \left( {g - f} \right)\left( x \right) = g\left( x \right) - f\left( x \right)$

Therefore, putting $f\left( x \right) = 4 - {x^2}$ and $g\left( x \right) = 6x$ in the above formula, we can write,
$\Rightarrow \left( {g - f} \right)\left( x \right) = 6x - \left( {4 - {x^2}} \right)$
That gives,
$\Rightarrow \left( {g - f} \right)\left( x \right) = {x^2} + 6x - 4$
Now we can find the value of the function $\left( {g - f} \right)\left( 3 \right)$ by putting $x = 3$ in the above equation,
Putting $x = 3$ gives us,
$\Rightarrow \left( {g - f} \right)\left( 3 \right) = {3^2} + 6\left( 3 \right) - 4$
Solving the RHS, we get the equation as
$\Rightarrow \left( {g - f} \right)\left( 3 \right) = 9 + 18 - 4$
Hence,
$\therefore \left( {g - f} \right)\left( 3 \right) = 23$

Therefore, $\left( {g - f} \right)\left( 3 \right)$ is equivalent to $23$.

Note: There is another property of functions which is known as the composition of functions. Function composition is the combination of two functions to form a new function. The composition of two functions $g$ and $f$ is the new function we get by performing $f$ first, and then performing $g$ . Composition of a function is done by substituting one function into the other function. One simply takes the output of the first function and uses it as the input to the second function. Mathematically,
$\Rightarrow \left( {fog} \right)\left( x \right) = f\left( {g\left( x \right)} \right)$