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. $$h(x, y)=(x+y) /(x-y).$$

Short answer

Question: Determine the domain and range of the function $$h(x, y) = \frac{x + y}{x - y}$$ and describe the graph of the function. Answer: The domain of the function $$h(x, y) = \frac{x + y}{x - y}$$ is $$(x, y) \in \mathbb{R}^2 : x \neq y$$, and its range is $$h(x, y) \in \mathbb{R}$$. The graph of the function exhibits a hyperbolic behavior, with a discontinuity along the diagonal line $$x = y$$.

Step-by-step solution

Find the domain

To find the domain of the function $$h(x, y) = \frac{x + y}{x - y}$$, we need to determine the values of $$x$$ and $$y$$ for which the function is defined. The only situation where this function is not defined is if the denominator $$x - y$$ is equal to 0, as division by 0 is not allowed. So, to find the domain, we need to determine when $$x - y \neq 0$$. Let's solve for when the denominator is equal to 0: $$x - y = 0$$ $$x = y$$ The function is not defined when $$x = y$$. Hence, the domain of $$h(x, y)$$ is all real numbers except when $$x = y$$, or we can say it as: Domain: $$(x, y) \in \mathbb{R}^2 : x \neq y$$

Find the range

To find the range of $$h(x, y)$$, we need to determine the possible values this function can take. The function is a fraction of two linear expressions, with the only restrictions coming from the denominator. Assuming the denominator $$x - y \neq 0$$, there are no other restrictions on the function's values, so any real value can be obtained by $$h(x, y)$$. Hence, the range of $$h(x, y)$$ is all real numbers: Range: $$h(x, y) \in \mathbb{R}$$

Graph the function

Graphing a multivariable function can be done using a graphing utility such as GeoGebra, Desmos, or a graphing calculator. Follow these steps to graph the function and experiment with different windows and orientations: 1. Open a graphing utility of your choice. 2. Input the function $$h(x, y) = \frac{x + y}{x - y}$$. 3. Experiment with different windows and orientations to obtain the best perspective of the surface. Ensure the graph correctly represents the domain and range found in previous steps. By doing so, you should observe that the graph of the function exhibits a hyperbolic behavior, with the diagonal line $$x = y$$ representing a discontinuity where the function is not defined. Overall, the domain of the function $$h(x, y) = \frac{x + y}{x - y}$$ is $$(x, y) \in \mathbb{R}^2 : x \neq y$$ and its range is $$h(x, y) \in \mathbb{R}$$.

Key concepts

Graphing Multivariable Functions

Graphing multivariable functions requires a bit of imagination, as these functions offer a view into a multi-dimensional space. In our function \( h(x, y) = \frac{x + y}{x - y} \), the variables \(x\) and \(y\) form input pairs whose output is a third dimension, often visualized as a surface. Graphing this function involves using a tool such as GeoGebra or Desmos, which can render three-dimensional plots. These tools help by offering different perspectives by rotating and adjusting the viewing angle. When plotting \( h(x, y) \), remember:

• Identify the axes: Typically, \(x\) and \(y\) are horizontal and vertical on the plane, while the resulting \(h(x, y)\) values provide the height or depth.
• Observe the surface: Pay close attention to critical features like asymptotes or holes, which indicate points where the function is undefined.
• Adjust the window: Zooming in and out or altering axis limits may help you notice trends such as slopes or peaks.

In this particular function, you'll find areas where it seems the surface splits or disappears, especially along the line \( x = y \). This is due to the discontinuity, which we'll explore further.

Discontinuity in Functions

Discontinuity in functions refers to the points where a function does not smoothly form a curve or a surface. In mathematical terms, a function is discontinuous at any point where it is not defined or does not have a real output.In the case of our function \( h(x, y) = \frac{x + y}{x - y} \), discontinuity occurs along the line \( x = y \). This is because, mathematically, a denominator of zero in rational functions creates a division-by-zero situation, which is undefined. As a result, there is a gap or break at \( x = y \) on the graph of this function. Here’s what these discontinuities often mean:

• The function's output suddenly shifts to infinity or some undefined state at \( x = y \).
• Graphically, it might mean a vertical asymptote, where the graph shoots up or down very steeply.
• Continuity is maintained anywhere else on the surface where \( x eq y \).

Recognizing discontinuities is crucial for understanding the behavior of complex functions, especially when working with their graphical representations or applying them to real-world problems.

Range of a Rational Function

The range of a function relates to all the possible output or "z" values the function can generate given every permissible input pair \( (x, y) \). For rational functions like \( h(x, y) = \frac{x + y}{x - y} \), the range is closely tied to the behavior of the function’s numerator and denominator. Since the function’s denominator dictates where it is undefined, the critical task is assessing the outputs where the function does resolve. In our case:

• Assuming \( x eq y \) (thus avoiding the discontinuity), the numerator \( x + y \) can vary freely across real numbers.
• This flexibility means that as \( x \) and \( y \) change, \( h(x, y) \) can extend to nearly any real number.
• Consequently, the range of \( h(x, y) \) is all real numbers \( h(x, y) \in \mathbb{R} \).

Understanding the range helps further plot the function correctly and indicates possible values your function might take in practical applications. It is a reassuring aspect of rational functions, as they often cater to a vast array of potential results.