Largest subset of the reals, on which a function can be defined

Question

I was just wondering what it means by 'write down the largest subset of the reals, of which the function is defined' and the function in this case is a composition function so for example $(f \circ g)(x)$. Any help is appreciated. Thanks in advance.

Answer 1

For a real number $x \in \mathbb{R}$ to be defined in the composition function $(f\circ g)(x)$, $x$ must be defined in $g$, and $g$ must map it to a defined value in $f$.

So the answer would be the domain of $g$ which maps to the domain of $f$. In other words, let $I_g$ be the image of $g$, and $D_f$ be the domain of $f$. For some $x \in \mathbb{R}$, $x$ is defined in the composition function if and only if

$$g(x) \in D_f \Rightarrow x \in g^{-1}(D_f) \Rightarrow x\in g^{-1}(I_g\cap D_f)$$

Hope this helps!