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Question: The composite mapping \(fog\) of the map \(f:R \to R,f\left( x \right) = \sin x\) and \(g:R \to R,g\...

The composite mapping fogfog of the map f:R→R,f(x)=sin⁡xf:R \to R,f\left( x \right) = \sin x and g:R→R,g(x)=x2g:R \to R,g\left( x \right) = {x^2} is:
A. x2sin⁡x{x^2}\sin x
B. (sin⁡x)2{\left( {\sin x} \right)^2}
C. sin⁡x2\sin {x^2}
D. sin⁡xx2\dfrac{{\sin x}}{{{x^2}}}

Explanation

Solution

We have to calculate the value of the composite equation, fogfog, when we are given f:R→R,f(x)=sin⁡xf:R \to R,f\left( x \right) = \sin x and g:R→R,g(x)=x2g:R \to R,g\left( x \right) = {x^2}. We will substitute the value of g(x)g\left( x \right) and then find the value of f(g(x))f\left( {g\left( x \right)} \right)

Complete step by step solution:
We are given that f:R→Rf:R \to R and f(x)=sin⁡xf\left( x \right) = \sin x.
The function g:R→Rg:R \to R and g(x)=x2g\left( x \right) = {x^2}
We know that fogfog is given as f(g(x))f\left( {g\left( x \right)} \right)
Substitute the value of g(x)=x2g\left( x \right) = {x^2} in the above expression.
f(x2)f\left( {{x^2}} \right)
Since, f:R→Rf:R \to R and f(x)=sin⁡xf\left( x \right) = \sin x
So, f(x2)=sin⁡(x2)f\left( {{x^2}} \right) = \sin \left( {{x^2}} \right)
Hence, the value of fog(x)fog\left( x \right) is sin⁡(x2)\sin \left( {{x^2}} \right)

Thus, option C is correct.

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