Finding the Domain of a Composite Function

The domain of $g\left(x\right)$ consists of all real numbers except $x=\frac{2}{3},$ since that input value would cause us to divide by 0. Likewise, the domain of $f$ consists of all real numbers except 1. So we need to exclude from the domain of $g\left(x\right)$ that value of $x$ for which $g\left(x\right)=1.$

So the domain of $f\circ g$ is the set of all real numbers except $\frac{2}{3}$ and $2.$ This means that

We can write this in interval notation as