g(x)= 6/x find (g°g). and domain in set notation.

that is we firsts evaluate the inner functions that in this case is g, now taking as argument y=6/x, we evaluate the outer function that in this case also is g, as follows:

[tex]g\text{ \lparen }\frac{6}{x})=\frac{6}{\frac{6}{x}}=\frac{6}{6}=x[/tex]

That is, the composition g*g is equal to x, the identity.

Now we will find the domain of g*g:

Note that the domain of a composition is an interception, as follows:

[tex]Domain\text{ }g\circ\text{ g=\textbraceleft Domain of }g\text{ \textbraceright }\cap\text{ \textbraceleft Image of }g\text{ \textbraceright}[/tex]

Therefore, we have to find the domain and image of g, and intercept both sets. We start with the domain of g_

[tex]Domain\text{ of }g\text{ }=\text{ }\mathbb{R}\text{ - \textbraceleft0\textbraceright}[/tex]

That is all the real numbers except the 0. Now note that the image of g is

[tex]Image\text{ g= }\mathbb{R}\text{ - \textbraceleft0\textbraceright}[/tex]

Finally, the domain of the composition g*g, can be obtained by the formula above:

[tex]Domain\text{ of }g\circ\text{ g=}\mathbb{R}\text{ -\textbraceleft0\textbraceright }\cap\text{ }\mathbb{R}\text{ - \textbraceleft0\textbraceright= }\mathbb{R}\text{ - \textbraceleft0\textbraceright=}(-\infty\text{ },0)\text{ }\cup\text{ }(0,\infty)\text{ }[/tex]

Therefore, the domain of the composition are all the real numbers excluding the 0.

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