Question

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function, and find the range of the function. $$ f(x, y)=x^{2}+y^{2}-1 $$

Short answer

The domain of the function \(f(x, y) = x^{2}+y^{2}-1\) is \(R^{2}\), which indicates all pairs of real numbers. The range of this function is \([-1, +\infty)\], as the function's minimum value can be -1, and it can generate any value greater than or equal to -1.

Step-by-step solution

Determine the Domain

For any real-valued function, the domain constitutes all the real numbers for which the function is defined. In this case, the function \(f(x, y) = x^{2}+y^{2}-1\) is defined for all pairs of real numbers \((x, y)\). Therefore, the domain of this function is \(R^{2}\), which signifies all real numbers as a 2-dimensional plane.

Determine the Range

To determine the range of the function, we need to analyze the possible outputs of the function. The minimum value of the function \(f(x, y)=x^{2}+y^{2}-1\) occurs when \(x^{2} + y^{2}\) is minimized. Since squares of real numbers are non-negative, the minimum value of \(x^{2} + y^{2}\) will be 0 when both x and y are 0, making the minimum of the function -1. The function will take any value above or equal to this minimum. So, the range of this function is \([-1, +\infty)\].

Key concepts

Real-Valued Function

When examining functions in mathematics, it's important to distinguish between different types of functions. A real-valued function is a function that provides a real number as an output for each input from its domain. The concept of a real-valied function is foundational in understanding the behavior of equations that model real-world situations.

For instance, the exercise involves the function f(x, y) = x^2 + y^2 - 1. This function takes two real numbers as its input and produces a single real number as its output. Essential to note, the output here represents real conditions or quantities, which could be distances, areas, or other measurable attributes in a real-world scenario. This is distinct from complex-valued functions, which might produce outputs involving complex numbers - numbers that include the square root of negative one. In our daily experiences and scientific measurements, we majorly deal with real numbers, making real-valued functions extremely relevant and practical.

2-Dimensional Plane

In the world of mathematics, especially when dealing with functions of two variables, we often represent the domain on a 2-dimensional plane. This plane is an expanse of points where each point can be identified using a pair of real numbers. These pairs of numbers represent the coordinates of a point on the plane – the x (horizontal) and y (vertical) axes.

The domain of our example function, f(x, y) = x^2 + y^2 - 1, consists of all pairs of real numbers that can be plotted on this plane. Imagine an infinite flat surface extending in all directions; any point you can mark represents a potential input for our function. This visual can be highly beneficial in understanding the behavior of functions as it allows us to visualize concepts such as domain and range easily.

Function Minimum Value

Identifying the minimum value of a function is crucial when analyzing its range. The minimum value is the lowest point on the graph of the function when we visualize it on the 2-dimensional plane. This value tells us the least output the function can produce.

With respect to our function, f(x, y) = x^2 + y^2 - 1, locating the minimum value involves finding the smallest result the function can yield. Given that both x^2 and y^2 are always non-negative (since squaring a real number never gives a negative result), the minimum sum of x^2 + y^2 is when x and y are both zero. Hence, the function's minimum value occurs at this point, which is -1 in this case. The significance of this minimum value is profound as it anchors the lower bound of the range of the function – meaning our function's output will never dip below this minimum value.