Question

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

Answer 1

In this question, they have given the value of given function $f(x)$ and $g(x)$ , and asked us to find the value of $\left( {f - g} \right)\left( 4 \right)$ . As we know, $(f - g) = f(x) - g(x)$ , first we need to subtract $g(x)$ from $f(x)$ and then substitute number $4$ in the place of $x$ of the resultant term or expression to get the required answer.

Formula used:
$(f - g)(x) = f(x) - g(x)$

Complete Step by Step Solution:
Here, they have given the value of a given function $f(x)$ and $g(x)$ , and asked us to find the value of $\left( {f - g} \right)\left( 4 \right)$ .
First we need to find the value of $\left( {f - g} \right)(x)$ and then substitute the number $4$ in the place of $x$ in it.
We know that, according to the identity of the functions,
$(f - g)(x) = f(x) - g(x)$
Therefore we need to obtain $f(x) - g(x)$
Here,
$f\left( x \right) = 4x - 3$
$g(x) = {x^3} + 2x$
Substituting the values we get,
$f(x) - g(x) = (4x - 3) - ({x^3} + 2x)$
Multiplying the minus inside the bracket, the signs will get changed.
$f(x) - g(x) = 4x - 3 - {x^3} - 2x$
Rearranging the equation,
$f(x) - g(x) = 4x - 2x - 3 - {x^3}$
And it becomes,
= $2x - 3 - {x^3}$
This is the value of $f(x) - g(x)$ .
Now, to evaluate $(f - g)\left( 4 \right)$ we need to substitute $x = 4$ into $(f - g)(x)$
Substituting $x = 4$ in $(f - g)(x)$ , we get
$(f - g)(4) = (2 \times 4) - 3 - {(4)^3}$
$= 8 - 3 - 64$
$(f - g)(4) = - 59$

Therefore the value of $\left( {f - g} \right)\left( 4 \right)$ is $- 59$

Note: The concept and understanding of functions are easy. The notation $y = f\left( x \right)$ defines a function named$\;f$ . This is read as “$y$ is a function of $x$ .” Here the letter $x$ represents the input value, or independent variable. The letter $y$ , or$f\left( x \right)$, represents the output value, or dependent variable.
The property of limits of function:
The two functions can be added, subtracted, multiplied and divided.
It is given that,
$(f + g)(x) = f(x) + g(x)$
$(f - g)(x) = f(x) - g(x)$
$(f \times g)(x) = f(x) \times g(x)$
$(f \div g)(x) = f(x) \div g(x)$
Also, If f and g are two functions and both $li{m_{x \to a}}\;f\left( x \right)$ and $li{m_{x \to a}}\;g\left( x \right)$ exist, then
The limit of the sum of two functions is the sum of their limits.
$lim\left[ {f\left( x \right) + g\left( x \right)} \right] = lim{\text{ }}f\left( x \right) + lim{\text{ }}g\left( x \right)$
The limit of the difference of two functions is the difference of their limits.
$lim\left[ {f\left( x \right) - g\left( x \right)} \right] = lim{\text{ }}f\left( x \right) - lim{\text{ }}g\left( x \right)$
The limit of the product of two functions is the product of their limits.
$lim\left[ {f\left( x \right) \times g\left( x \right)} \right] = lim{\text{ }}f\left( x \right) \times lim{\text{ }}g\left( x \right)$
The limit of the quotient of two functions is the quotient of their limits if the limit in the denominator is not equal to 0.
$lim\left[ {f\left( x \right) \div g\left( x \right)} \right] = lim{\text{ }}f\left( x \right) \div lim{\text{ }}g\left( x \right)$; If $lim{\text{ }}g\left( x \right)$ is not equal to zero.