In function spaces, we sometimes define the dot product for two elements $f_1, f_2$ as
$$ (f_1, f_2) \equiv \int_a^b f_1^*(x)f_2(x)\rho(x) \mathrm{d}x $$
where $\rho(x)$ is the weight function and $f_1^*(x)$ is the conjugate of $f_1(x)$.
There are two things that I do not understand.
Why do we need conjugates here? Why are they mathematically necessary? I have made little connections between modules in complex space and function spaces, in both we have the multiplication of the conjugated pair:
$$ \lambda = (a+\mathrm{i} b) \implies \lambda^* = (a-\mathrm{i}b) $$
such that
$$ \|\lambda\|^2 = \lambda^*\lambda\implies \|\lambda\| = \sqrt{(a+\mathrm{i}b)(a-\mathrm{i}b)} = \sqrt{a^2+b^2} $$
What do conjugate functions look like? For example, what is the conjugate function of $f(x) = 3x^2 +5$ ? Is it associated with imaginary numbers or is conjugate just an idea?