Question

If \[f\] be the greatest integer function defined as and \[g\] be the modulus function defined as \[g(x) = |x|\], then the value of \[gof\left( {\dfrac{{ - 5}}{4}} \right)\] is
A. \[\dfrac{1}{2}\]
B. \[0\]
C. \[2\]
D. Does not exist

Answer 1

Hint: In this question we have to find the value of \[gof\left( {\dfrac{{ - 5}}{4}} \right)\]. This is the composition of two functions. We first find the value of the inner function putting in \[x = \left( {\dfrac{{ - 5}}{4}} \right)\] in the \[f(x) = [x]\]. Now we put the resultant value of inner function into the function \[g(x) = |x|\] and solve to get the desired result.

Complete step-by-step answer:
The given question is based on the composition of two functions.
Consider the given question,
The given function are \[f(x) = [x]\] and \[g(x) = |x|\].
Now we know that \[gof(x) = g(f(x))\]
Taking \[x = \left( {\dfrac{{ - 5}}{4}} \right)\], we get
\[gof\left( {\dfrac{{ - 5}}{4}} \right) = g\left( {f\left( {\dfrac{{ - 5}}{4}} \right)} \right)\]
Now we find the value of \[f\left( {\dfrac{{ - 5}}{4}} \right)\].
Hence, we have \[f(x) = [x]\], where [ ] denotes the greatest integer function .
Taking \[x = \left( {\dfrac{{ - 5}}{4}} \right)\], we get
\[f\left( {\dfrac{{ - 5}}{4}} \right) = \left[ {\left( {\dfrac{{ - 5}}{4}} \right)} \right]\]
Since the value \[\left( {\dfrac{{ - 5}}{4}} \right) = \left( { - 2 + \dfrac{3}{4}} \right)\] lies between \[ - 2\] and \[ - 1\] .
Therefore \[f\left( {\dfrac{{ - 5}}{4}} \right) = \left[ {\left( {\dfrac{{ - 5}}{4}} \right)} \right] = - 2\].
Hence, \[gof\left( {\dfrac{{ - 5}}{4}} \right) = g\left( {f\left( {\dfrac{{ - 5}}{4}} \right)} \right) = g\left( { - 2} \right)\]
Now we find the value of \[g\left( { - 2} \right)\].
Hence, we have \[g(x) = |x|\], where \[|\;\;|\] denotes the modulus function that gives positive values of the given number.
Taking \[x = \left( { - 2} \right)\], we get
\[g\left( { - 2} \right) = | - 2| = 2\]
Therefore, \[gof\left( {\dfrac{{ - 5}}{4}} \right) = g\left( {f\left( {\dfrac{{ - 5}}{4}} \right)} \right) = 2\]
Hence Option \[(C)\]is correct.
So, the correct answer is “Option C”.

Note: The greatest integer function is the function that gives an integer value. Any number can be written as a sum of integers and fractions. Thus the greatest integer function gives the integer part of the number. For example: the greatest integer function \[[1.25]\] is \[1\].
i.e. \[[1.25] = [1 + 0.25] = 1\]
Similarly we can find the value of any GIF.
Modulus function always gives the positive value of any number.
For example: \[| - 5| = 5\].
The value of composition of function when order of composition is reversed is not always equal. i.e. \[gof(x) \ne fog(x)\].