How to define a restriction on composite functions

[violet plaza]

You see the input of f(x) is limited from -inf to 3. But since g(x) has output for all real numbers, including 3 to inf, f(g(x)) is not defined.
The restriction you can include here would be such that g*(x) is a valid input for f(x)
||R -> (-inf, 3]||

[sharp flume]

huh 😭

no thats not the anserr 💔

[violet plaza]

wut is the answer?

[stable spear]

f(g(x)) = -x² + 4

but since x² - 1 has a range of [-1,oo) and f(x) has a domain of [-oo, 3]

So x² - 1 should have a range of [-1,3] thus x ∈ [-2;2] is the domain of f(g(x))

[violet plaza]

of course it is @stable spear with the clutch 🛐

[stable spear]

You can also use this definition:
Suppose:

$\g : A \rightarrow B$ and $f : C \rightarrow D\$

$\ f \circ g = f(g(x)) : {x \in A | g(x) \in C}$

[crystal owl]

adonhs