Operations with Rationals Precal homework domain problem

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Find $(f ∘ g)(x)$ and state the domain: $f(x) = (2x+1)/(x-3)$ and $g(x)=1/(x-1)$

By plugging in $1/(x-1)$ in the x of f(x) I ot $(x+1)/(-3x+4)$, and since the domain of g(x) was the original domain, x does not equal 1 but because of $(-3x+4)$, x also does not equal $4/3$. However, in the answer, they only state that x does not equal 1. What went wrong?

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