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## f(x)= square root of 2x^2-1

what is the domain of this function √2x2-1

Do you mean f(x) = √(2x2-1)?
If yes, then the domain requires that 2x2-1 ≥ 0. Solving the inequality gives the domain [-1/√2, 1/√2]
So the domain is a set of all possible inputs, and the range is a set of all possible outputs.

To determine if your inputs are limited try to find a value of x which would result in an undefined value.

Is your function f(x) = sqr(2x2-1) or f(x) = sqr(2x2) - 1 ?  because it can make a significant difference.

If we assume we are only dealing with real numbers, then an expression which would give you the square root of a negative number would be undefined, and the value of x which causes this would not be in the domain.

Remember not to restrict your thinking about the values of x to be only integers. Think of the consequences of fractional values as well.

Consider all the values of the domain and which values of f(x) you could get. That would define your range.