Question

a. Determine the domain and range of the following functions. b. Graph each function using a graphing utility. Be sure to experiment with the window and orientation to give the best perspective of the surface. $$f(x, y)=|x y|.$$

Short answer

Answer: The domain of the function $$f(x,y)=|xy|$$ is all real numbers for both x and y, denoted as $$(x,y) \in \mathbb{R}^2$$. The range of the function is all non-negative real numbers, as $$f(x,y) \geq 0$$. In a 3D graph, the surface would open upward and be symmetrical along the z-axis, with non-negative values for z-coordinates.

Step-by-step solution

Determine the domain of the function

To find the domain of $$f(x,y)=|xy|$$, we need to identify the values of x and y for which the function is defined. The function contains the absolute value of the product of x and y, so it is defined for all values of x and y. Thus, the domain of the function is all real numbers, which can be written as: $$Domain: (x,y) \in \mathbb{R}^2$$.

Determine the range of the function

To find the range of $$f(x,y)=|xy|$$, we need to identify the possible output values of the function given any valid input values in its domain. Since the product of any two real numbers is also a real number, xy is a real number. Then taking the absolute value will always result in a non-negative value. So, the range of the function is all non-negative real numbers, which can be written as: $$Range: f(x,y) \geq 0$$.

Graph the function using a graphing utility

As we cannot graph the function directly here, below are the steps to graph the function using a graphing utility: 1. Open a graphing utility such as GeoGebra or Desmos. 2. Input the function $$f(x, y) = |xy|.$$ The graphing utility should be able to handle 3D graphing for this exercise. 3. Adjust the window and orientation to provide the best perspective of the surface. You might need to zoom in or out and rotate the graph to find the best view. 4. The graph of the function should appear as a surface in 3D space, with the z-axis representing the function's output values. The function will be symmetrical along the z-axis because of the absolute value operation. The graph of the function $$f(x,y)=|xy|$$ should display a surface that opens upward and is symmetrical along the z-axis. The output values (z-coordinates) will always be non-negative, verifying our previous calculation of the range.

Key concepts

Domain and Range

Understanding the domain and range of a multivariable function is a fundamental concept in multivariable calculus. When dealing with the function \(f(x,y) = |xy|\), the domain refers to all the possible input values for \(x\) and \(y\) where the function is well-defined. Since \(|xy|\) involves multiplication and absolute value, there aren't any restrictions like division by zero or taking square roots of negative numbers. Thus, both \(x\) and \(y\) can be any real number.

• Domain: All real numbers, \((x, y) \in \mathbb{R}^2\).

Next, let's consider the range of \(f(x,y) = |xy|\). The range denotes all possible output values. The multiplication \(xy\) results in a real number, and applying the absolute value makes sure that the result is always non-negative. Hence, the outputs are non-negative real numbers.

• Range: All non-negative real numbers, \(f(x,y) \geq 0\).

Graphing Functions

Graphing a multivariable function like \(f(x,y) = |xy|\) can vividly illustrate its behavior and characteristics. It’s essential to use graphing utilities capable of 3D graphing, such as GeoGebra or Desmos, to achieve a clear visualization. Initially, input the function into the graphing tool. The tool will generate the graph, which will appear as a 3D surface. It may sometimes be necessary to adjust the viewing window and orientation for clarity. Proper configuration ensures that you capture the symmetry and overall shape of the graph.

The surface you generate with \(f(x,y) = |xy|\) will open upwards, forming a shape similar to a paraboloid.

It will be symmetrical along the z-axis, reflecting the absolute value nature of the function, implying symmetry in output values for hidden negatives.

3D Coordinate Systems

Understanding the role of 3D coordinate systems is crucial when graphing multivariable functions. In 3D graphing, we deal with three axes: the \(x\)-axis, the \(y\)-axis, and the \(z\)-axis. Each axis represents a different dimension in space. When graphing \(f(x,y) = |xy|\), the \(x\)- and \(y\)-axes correspond to the input variables, while the \(z\)-axis represents the function's output (or response) values. The point \((x, y, f(x,y))\) denotes a specific location on the graph's surface.

• The interaction between the input axes \((x, y)\) determines the shape and position of the resulting surface.
• As you navigate through different inputs, you trace out the surface within the 3D space.
• Observing how changes in \(x\) and \(y\) affect \(z\) provides insight into the function's behavior.

Being adept with 3D coordinate systems allows you to better understand and visualize the interaction of multiple variables within the function \(f(x,y) = |xy|\). Explore different perspectives by rotating the 3D plot to gain deeper insights into the symmetry and shape of the surface.