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How do I make functional substitutions?

I want to substitute one function for another. Consider this:

v = function('x')
u(x) = (v(x)+v(-x))/2
delta = 10^(-90)

The very last line is what I'm trying to accomplish. I know it's the wrong command, but I'm wanting to make the substitution $v(x)\to\Gamma(x)$.

Is there a way along these lines to do such a functional substitution?