Question

The composite mapping $fog$ of the map $f:R \to R,f\left( x \right) = \sin x$ and $g:R \to R,g\left( x \right) = {x^2}$ is:
A. ${x^2}\sin x$
B. ${\left( {\sin x} \right)^2}$
C. $\sin {x^2}$
D. $\dfrac{{\sin x}}{{{x^2}}}$

Answer 1

Hint:
We have to calculate the value of the composite equation, $fog$, when we are given $f:R \to R,f\left( x \right) = \sin x$ and $g:R \to R,g\left( x \right) = {x^2}$. We will substitute the value of $g\left( x \right)$ and then find the value of $f\left( {g\left( x \right)} \right)$

Complete step by step solution:
We are given that \[f:R \to R\] and $f\left( x \right) = \sin x$.
The function $g:R \to R$ and $g\left( x \right) = {x^2}$
We know that $fog$ is given as $f\left( {g\left( x \right)} \right)$
Substitute the value of $g\left( x \right) = {x^2}$ in the above expression.
$f\left( {{x^2}} \right)$
Since, \[f:R \to R\] and $f\left( x \right) = \sin x$
So, $f\left( {{x^2}} \right) = \sin \left( {{x^2}} \right)$
Hence, the value of $fog\left( x \right)$ is $\sin \left( {{x^2}} \right)$

Thus, option C is correct.

Note:
We need to know that $fog\left( x \right)$ is equal to $gof\left( x \right)$. If we will calculate $gof\left( x \right)$, we will first the value of $f\left( x \right)$, then \[g\left( {\sin x} \right)\] is ${\left( {\sin x} \right)^2}$ and it not equal to $fog\left( x \right)$