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

Hint: 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.