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To perform function analysis on a multivariable function like \(f(x, y) = x^2 + y^2 - 1\), we start by understanding the formulation and properties of the function. Here, the function comprises squares of two variables, \(x\) and \(y\), which are always non-negative due to the squaring operation.
Next, we reduce the problem to its simplest form by analyzing scenarios where the function may achieve its extremum values. Since the expression in the function is composed of two non-negative squares, the function's minimum occurs when both \(x\) and \(y\) are zero, giving \(f(0, 0) = 0^2 + 0^2 - 1 = -1\).
Recognizing that squares \(x^2\) and \(y^2\) have no upper bound, the output value of this function can increase without limit. This implies the absence of a maximum value for \(f(x,y)\). This analysis helps identify the range of the function, which is crucial for understanding what output values are possible.

The range of a function refers to the set of all output values it can produce. In this context, for the function \(f(x, y) = x^2 + y^2 - 1\), the range is determined by calculating the possible minimum and understanding the behavior of outputs as \(x\) and \(y\) vary.

• Finding the Minimum: As previously analyzed, the minimum value of this function is \(-1\), achieved when \(x = 0\) and \(y = 0\). This is because the expression \(x^2 + y^2\) cannot result in a negative number.
• Understanding the Maximum: There is no upper bound for \(f(x, y)\) as the values of \(x^2 + y^2\) can increase indefinitely with increasing \(x\) and \(y\). Therefore, procedural squaring allows this function to reach output values towards infinity.

The combination of these insights leads us to conclude that the range is \([-1, +\infty)\). This means any real number greater than or equal to \(-1\) can be an output of the function.

Understanding the domain of a function is vital to comprehend over what input values the function operates. For a function with real numbers, the domain consists of all possible ordered pairs \((x, y)\) where each of \(x\) and \(y\) are real numbers in the set \(\mathbb{R}\). This implies that \(f(x, y) = x^2 + y^2 - 1\) covers the entire \(xy\)-plane.

• Real Numbers: These are numbers that include both rational and irrational numbers, meaning \(x\) and \(y\) can take any value within this set, such as integers, fractions, or even non-repeating decimals.
• Indefinite Extent: In this specific function, there are no constraints like a square root or a division by zero that would limit \(x\) or \(y\). Hence, every real number is valid for both variables, allowing any combination of \(x\) and \(y\).

Therefore, the domain of this function is all real numbers \((x, y)\in \mathbb{R}^2\). It is important to know the domain as it determines the complete set of possible inputs to the function.