Are Two Functions Equal?
Question:
Let
$$
f:\mathbb{R}\to\mathbb{R}
$$
and
$$
g:\mathbb{C}\to\mathbb{C}
$$
be two functions defined as
$$
f(x)=x^2
$$
and
$$
g(x)=x^2
$$
Are they equal functions?
Solution
Two functions are equal if they have:
- same domain
- same codomain
- same rule of correspondence
Here,
$$ f:\mathbb{R}\to\mathbb{R} $$
and
$$ g:\mathbb{C}\to\mathbb{C} $$
Although both functions have the same rule
$$ x\mapsto x^2 $$
their domains and codomains are different.
Function \(f\) is defined on real numbers, whereas function \(g\) is defined on complex numbers.
Therefore, the two functions are not equal.
$$ \boxed{f\ne g} $$