Question

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

Answer 1

We are given two functions f(x) and g(x) and we have to find the difference of these two functions. And we have to find a difference for a particular value of x. For this first we will find the difference between f(x) and g(x) then we will substitute the value in place of x of which we have to find a value. This procedure will find the value.

Complete step-by-step answer:
Step1: We are given two function i.e. $f\left( x \right) = 4x - 3$ and $g\left( x \right) = {x^3} + 2x$ and we have to find $\left( {f - g} \right)\left( 4 \right)$. So for this first we will find the difference between $f\left( x \right)$ and $g\left( x \right)$ so we will subtract $g\left( x \right)$ from $f\left( x \right)$.
$\Rightarrow \left( {f - g} \right)\left( x \right) = f\left( x \right) - g\left( x \right)$
On substituting the values of $f\left( x \right)$and$g\left( x \right)$we get:
$\Rightarrow f\left( x \right) - g\left( x \right) = 4x - 3 - \left( {{x^3} + 2x} \right)$
$\Rightarrow f\left( x \right) - g\left( x \right) = 4x - 3 - {x^3} - 2x$
On rearrangement we will get:
$\Rightarrow f\left( x \right) - g\left( x \right) = - {x^3} + 2x - 3$…. (1)
Step2: As we are given $\left( {f - g} \right)\left( 4 \right)$ so we will substitute the value for x as $x = 4$. On substituting the values we will get:
$\Rightarrow f\left( x \right) - g\left( x \right) = - {\left( 4 \right)^3} + 2\left( 4 \right) - 3$
On further solving we will get:
$\Rightarrow f\left( x \right) - g\left( x \right) = 8 - 3 - 64$
By adding and subtracting we will get:
$\Rightarrow f\left( x \right) - g\left( x \right) = - 59$

Hence the value is $- 59$

Note:
In these types of questions students mainly make mistakes in finding $\left( {f - g} \right)\left( x \right)$ they get confused about how to find. But it’s simply the difference. Just find the difference between the two functions and substitute the value of x of which we have to find. These types of questions are mainly solved by the instructions given. For example $\left( {f + g} \right)\left( x \right)$ here we will find the sum. $fog\left( x \right)$ In this case we have to find the product. Like this we can solve these types of questions according to the instructions given. In case of modulus function or greatest integer function and to solve $fog\left( x \right)$ we have to consider cases. But in these simple equations we have to solve them simply no conditions or cases are required.