find the domain of the composite function f . g f(x) =1/x-5 ; g(x) =  6/x

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lfryerda's profile pic

lfryerda | High School Teacher | (Level 2) Educator

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To find the domain of the composite function `f circ g(x)` where `f(x)=1/{x-5}` and `g(x)=6/x`, we need the intersection of the domains of each of the two functions `f circ g(x)` and `g(x)`.  That is, we need to remove any trouble spots from `f circ g(x)` and from the argument to `f(x)`, which is `g(x)`.

Since `f circ g(x)=1/{6/x-5}=x/{6-5x}`, this means that one place where the function is undefined is `x=6/5` since the denominator vanishes there, so that has to be removed from the domain.  However, we also need to look at where `g(x)=6/x` is undefined.  This happens when its denominator vanishes.  That is when `x=0` so that also needs to be removed from the domain.

This means that the domain of the composite function is `{x\in R| x ne 0, x ne 6/5}`.

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You first need to remember how the composition of two functions works:

`(fog)(x) = f(g(x))`

Notice that you need to substitute g(x) for x in equation of f(x) such that:

`f(g(x)) = 1/(g(x) - 5) =gt f(g(x)) = 1/(6/x - 5) =gt f(g(x)) = x/(6-5x)`

You need to remember that the domain of the function consists of all values of x for the function exists. Notice that the equation of the function is a fraction, hence, all the zeroes of denominator makes the function impossible to exist, hence, you need to find what the zeroes of denominator are and you need to exclude these values from domain such that:

`6 - 5x = 0 =gt -5x = -6 =gt x = (-6)/(-5) =gt x = 6/5`

Hence, evaluating the domain of the function yields that `x in R - {6/5}.`

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