Domain is all real numbers, how can i prove specific range?

Question

is there a general way to prove a specific range (ex. (-2,2)) for a function with a domain of all real numbers?

Answer 1

No, you have to look at the function and "test" for limit values.

Some functions can only return values that are positive (for example, the square) even if the domain covers the whole line (positive and negative).

Others can only return a limited range, usually cyclical functions. For example: sin(x) will always return a value in the interval [-1, +1], even if x is from anywhere in the real numbers.

But, no magical recipe.

Sorry.

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If you can write the inverse function, then (maybe) this inverse function has special rules for domain. The range of the original function is the domain of the inverse function.

For example, the inverse function of "square" is "root" and we know that you cannot take the square root of a negative number.