Question

If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?

Answer 1

$$\begin{gathered} \text{We know that} \\ f:R\to \left[-1,1\right]\mathrm{and}g:R\to R \\ \text{Clearly, the range of}f\text{is a subset of the domain of}g\text{.} \\ gof:R\to R \\ \left(gof\right)\left(x\right)=g\left(f\left(x\right)\right) \\ =g\left(\sin x\right) \\ =2\sin x \end{gathered}$$

$$\begin{gathered} \text{Clearly, the range of}g\text{is a subset of the domain of}f\text{.} \\ fog:R\to R \\ \mathrm{So},\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right) \\ =f\left(2x\right) \\ =\sin \left(2x\right) \end{gathered}$$

Clearly, fog$\ne$gof