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 $f(x)$ .