Question

If the function \[f(x)={{x}^{2}}\] , \[g(x)=\tan x\] and \[h(x)=\log x\] then \[\left\{ ho\left( gof \right) \right\}\left( \sqrt{\dfrac{\pi }{4}} \right)\] .
a.0
b.1
c.\[\dfrac{1}{x}\]
d.\[\dfrac{1}{2}\log \dfrac{\pi }{4}\]

Answer 1

Hint: First of all, we need to find \[gof\] . For that we have to replace x by \[f(x)\]in the function \[g(x)=\tan x\] . After replacing we get, \[g\left\{ f\left( x \right) \right\}=\tan f\left( x \right)\] , where \[f(x)={{x}^{2}}\] . Then, replace x by
\[g\left\{ f\left( x \right) \right\}\] in the function \[h(x)=\log x\] . After replacing we get, \[h[g\left\{ f\left( x \right) \right\}]=\log g\left\{ f\left( x \right) \right\}\] , where \[g\left\{ f\left( x \right) \right\}=\tan f\left( x \right)\] . Now solve it further after putting the value of function \[f(x)={{x}^{2}}\] where \[x=\sqrt{\dfrac{\pi }{4}}\] .

Complete step-by-step answer:
According to the question, we have the value of the function
\[f(x)={{x}^{2}}\] ……………….(1)
\[g(x)=\tan x\] ………………..(2)
\[h(x)=\log x\] …………………………(3)
We have to find the value of \[\left\{ ho\left( gof \right) \right\}\left( \sqrt{\dfrac{\pi }{4}} \right)\] . For that, we need to find \[gof\] first.
For that we need to find \[gof\] that is \[gof=g\left\{ f\left( x \right) \right\}\] .
Replacing x \[f(x)\]in the function \[g(x)=\tan x\] , we get
\[gof=g\left\{ f\left( x \right) \right\}=\tan f(x)\] ………………(4)
From equation (1), we have \[f(x)={{x}^{2}}\] .
From equation (1) and equation (4), we get
\[gof=g\left\{ f\left( x \right) \right\}=\tan f(x)=\tan {{x}^{2}}\] …………………….(5)
Now, we are going to find \[\left\{ ho\left( gof \right) \right\}\] .
Replacing x by \[gof\] in the function \[h(x)=\log x\] , we get
\[\left\{ ho\left( gof \right) \right\}=h[g\left\{ f\left( x \right) \right\}]=\log g\left\{ f\left( x \right) \right\}\] ………………….(6)
From equation (5) we have \[g\left\{ f\left( x \right) \right\}=\tan {{x}^{2}}\] . Now, putting the value of \[g\left\{ f\left( x \right) \right\}\] in equation (6), we get
\[\left\{ ho\left( gof \right) \right\}=h[g\left\{ f\left( x \right) \right\}]=\log g\left\{ f\left( x \right) \right\}=\log \tan {{x}^{2}}\] ……………………..(7)
It is asked to find the value of \[\left\{ ho\left( gof \right) \right\}\left( \sqrt{\dfrac{\pi }{4}} \right)\] . So, put the value of x as \[\dfrac{\pi }{4}\] .
Now, putting \[x=\dfrac{\pi }{4}\] in equation (7), we get
\[\left\{ ho\left( gof \right) \right\}\left( \sqrt{\dfrac{\pi }{4}} \right)=\log \tan {{\left( \dfrac{\pi }{4} \right)}^{2}}\] ……………………………(8)
We know that, \[\tan \dfrac{\pi }{4}=1\] . Putting \[\tan \dfrac{\pi }{4}=1\] in equation (8), we get
 \[\left\{ ho\left( gof \right) \right\}\left( \sqrt{\dfrac{\pi }{4}} \right)=\log \tan {{\left( \dfrac{\pi }{4} \right)}^{2}}=\log {{1}^{2}}=\log 1=0\]
So, \[\left\{ ho\left( gof \right) \right\}\left( \sqrt{\dfrac{\pi }{4}} \right)=0\] .
Hence, the correct option is (A).

Note: In question one might make a mistake in finding \[gof\] . One might replace x by \[g(x)\] in the function \[f(x)\] which is wrong. Here order matters, if it is \[gof\] then we have to replace x by \[f(x)\] in the function \[g(x)\] and if it is \[fog\] then we have to replace x by \[g(x)\] in the function \[f(x)\] .