How to define a mathematical function in SymPy?

sympy.Function is for undefined functions. Like if f = Function('f') then f(x) remains unevaluated in expressions.

If you want an actual function (like if you do f(1) it evaluates x**2 + 1 at x=1, you can use a Python function

def f(x):
    return x**2 + 1

Then f(Symbol('x')) will give a symbolic x**2 + 1 and f(1) will give 2.

Or you can assign the expression to a variable

f = x**2 + 1

and use that. If you want to substitute x for a value, use subs, like

f.subs(x, 1)