How to substitute part of expression with function variable (Goal: create function that solves expression numerically for one of the variables)

Question

I have a complicated expression that contains N variables.

I want to create a function of N-1 that solves for one of those variables

Toy example:

expr = a^2-Sin[a*b]
astar[b_] := NSolve[expr==0, a, Reals]
astar[1]

does not work. I hoped it was equivalent to

NSolve[(expr/.{b->1})==0,a, Reals]

which does work.

I also tried

expr = a^2-Sin[a*b]
astar[b_] := NSolve[(expr/.{b->b})==0, a, Reals]
astar[1]

but that also doesn't work.

Answer 1

Functions should have explicit parameters.

expr[a_, b_] = a^2 - Sin[a*b];

Functions that use numeric techniques should restrict their parameters to numeric values.

astar[b_?NumericQ] := NSolve[expr[a, b] == 0, a, Reals];

astar[1]

(* {{a -> 0.}, {a -> 0.876726}} *)

Alternatively, get the exact solution

astar2[b_] := Solve[expr[a, b] == 0, a, Reals];

The exact solution is expressed as a Root expression

(* {{a -> 0}, 
   {a -> Root[{-Sin[#1] + #1^2 & , 
     0.8767262153950624459721623939201240\
      4608`20.315993684930827}]}} *)

Its approximate numeric value is the same as provided by astar

% // N

(* {{a -> 0.}, {a -> 0.876726}} *)