For a math assignment I was assigned by my professor, I've been asked to find the domain of a composition of the following functions.
$f(y) =\frac{4}{y - 2}$
$g(x) =\frac{5}{3x - 1}$
I know that the domain of $f(y)$ is all real numbers excluding $y = 2$ and that the domain of $g(x)$ is all real numbers excluding x = 1/3. As such, it makes sense to me that a composition of the two functions should be restricted in the same way, such that the input to $g(x)$ cannot be $1/3$ (because $g(1/3)$ is undefined) and cannot cause $g(x) = 2$ (because $f(2)$ is undefined). When trying to graph a composition of these two functions, however, I get a function that is defined at $1/3$ (desmos.com shows $f(g(1/3)) = 0)$. Can anyone help me wrap my head around this? Am I going about finding a composition the wrong way?
The domain of the composition $ f\circ g$ is $$D_{f(g)}=$$
$$\{x\in \Bbb R \;:\; x\in D_g \; and\; g(x)\in D_f\}$$
Or, in other equivalent way
Let $ x\in \Bbb R$. Then
$$x\in D_{f(g)}\iff $$ $$x\in D_g \; and \; g(x)\in D_f$$
So, in your example, as you said
$$x\in D_{f(g)}\iff $$ $$x\ne \frac 13\; and\; g(x)\ne 2$$
$$\iff x\ne \frac 13\; and \; \frac{5}{3x-1}\ne 2$$
$$\iff x\in \Bbb R\backslash \{\frac 13,\frac 76\}$$