<s>[INST] For $f(x)=\frac {1}{x}$ and $g(x)=\sqrt{x-4}$, find the domain of the composite function $g\circ f(x)$. 
For $f(x)=\frac {1}{x}$ and $g(x)=\sqrt{x-4}$, find the domain of the composite function $g\circ f(x)$.

My Attempt
Here, $$f(x)=\frac {1}{x}$$
$$g(x)=\sqrt{x-4}$$
Now,
$$g\circ f(x)=\sqrt{\frac {1-4x}{x}}$$
How can I proceed further? 
 [/INST] We have $$g\circ f(x)=\sqrt{f(x)-4}=\sqrt{\frac{1}{x}-4}$$
This function is defined when $x\neq 0$ because of the $\frac{1}{x}$ and when $\frac{1}{x}-4\geq 0$ because of the $\sqrt{\quad}$. Now, if $x<0$ then $\frac{1}{x}<0$ and so $\frac{1}{x}-4<0$. Hence, we need $x>0$. Finally if $x>0$, then 
$$\frac{1}{x}-4\geq 0 \iff \frac{1}{x}\geq 4 \iff 1 \geq 4x \iff \frac{1}{4}\geq x.$$
This finally shows that the domain of $g\circ f$ is $D=(0,\frac{1}{4}]$.
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