Question

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

Answer 1

We have to find the expression for the composition

`g\circ\text{ g\lparen x\rparen}`

Where

`g(x)=(6)/(x)`

And express its domain in set notation. We will start by finding the expression for the composition

`g\circ\text{ }g(x)=g(g(x))=g((6)/(x))`

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:

`g\text{ \lparen }(6)/(x))=(6)/((6)/(x))=(6)/(6)=x`

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:

`Domain\text{ }g\circ\text{ g=\textbraceleft Domain of }g\text{ \textbraceright }\cap\text{ \textbraceleft Image of }g\text{ \textbraceright}`

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

`Domain\text{ of }g\text{ }=\text{ }\mathbb{R}\text{ - \textbraceleft0\textbraceright}`

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

`Image\text{ g= }\mathbb{R}\text{ - \textbraceleft0\textbraceright}`

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

`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{ }`

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

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