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

Question:

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

Solution:

We know that

$f: R \rightarrow[-1,1]$ and $g: R \rightarrow R$

Clearly, the range of $f$ is a subset of the domain of $g$.

gof $: R \rightarrow R$

$(g o f)(x)=g(f(x))$

$=g(\sin x)$

$=2 \sin x$

Clearly, the range of $g$ is a subset of the domain of $f$.

$f o g: R \rightarrow R$

So, $(f o g)(x)=f(g(x))$

$=f(2 x)$

$=\sin (2 x)$

Clearly, fog">gof