Question

Help please quick question


`f=(x)/(x-1)`
the domain is 1
composite f of f is: f(f(x))
finding the domain of the composite through
`(x)/(x-1) =1\\x=x-1\\0=-1\\`
using a graphing calculator, I know that a hole is at (1,1), but can I figure that out without using a graphing calculator and through the domain, say 1=0. Can somebody please just quickly explain this to me.

Answer 1

Explanation:

What you are trying to do, it solve for the zeros of the rational function. To find the domain of composite function,

`(x)/(x - 1)`

Substitute x/(x-1) for every x you see.

`( (x)/(x - 1) )/( (x)/(x - 1) - 1) = (x(x \\ - 1))/(1) = x(x - 1)`

We know that the function domain can not be 1, this functions works for all reals, so our domain is

All reals except 1.

Answer 2

f(f(x)) = x, domain is x≠1

Explanation:

You want the domain of f(f(x)), given that f(x) = x/(x-1).

Domain

The domain excludes any value of x where the function is not defined.

The function f(x) = x/(x -1) is undefined at x=1, so that value is excluded from the domain of f(x).

The composite function f(f(x)) can be written as ...

`f(f(x)) = ((x)/(x-1))/((x)/(x-1)-1)=(x(x-1))/(1(x-1))=x`

This form shows there are factors of (x-1) in numerator and denominator that cancel. This is what creates the "hole" in the graph of f(f(x)) = x at (1, 1).

The domain of f(f(x)) is x≠1.

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