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It's known that we can easily define a function with optional value which is not related with other arguments, but my question is how can I define a function with an optional value which is really related with other variables.

For example

fun[a_, b_:g[a]] = a + g[a]

where g[a] = a^2 is a given function. I'd like to set the optional value of b to be g[a].

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ClearAll[fun, g]
g[x_] := x^2;
fun[a_, b_: g] := a + b[a]

fun[5]

30

fun[5, g]

30

fun[5, 2 # &]

15

fun[5, t]

5 + t[5]

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Use a pure function as the optional value of the 2nd variable.

f[a_, b_:(#^2 &)] := a + b[a]

f[42]

1806

f[42, Sqrt]

42 + Sqrt[42]

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