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 on the surface. $$F(x, y)=\tan ^{2}(x-y)$$

Short answer

Based on the given function, $$F(x, y) = \tan^2(x-y)$$, determine its domain and range. Domain: $D(F) = \{(x, y) \in \mathbb{R}^2 | x - y \ne (2n+1)\frac{\pi}{2}, n \in \mathbb{Z}\}$ Range: $R(F) = [0, +\infty)$

Step-by-step solution

Write down the function

The given function is: $$F(x, y) = \tan^2(x-y)$$

Analyze the domain of the function

In order to find the domain of the function, we have to identify the values of \(x\) and \(y\) for which the function is defined. The tangent function, \(\tan(x)\), is undefined whenever its argument is a multiple of \((2n+1)\frac{\pi}{2}\), where \(n\) is an integer. In this case, the argument of the tangent function is \((x-y)\). Hence, $$F(x, y)=\tan^2(x-y)$$ will be undefined whenever \(x - y = (2n+1)\frac{\pi}{2}\) for integer values of \(n\). In other words, the only values for which the function will not be defined are those pairs \((x, y)\) satisfying the condition \(x - y = (2n+1)\frac{\pi}{2}\).

Determine the domain of the function

Since the function is defined for almost all real values of \((x, y)\) except those pairs \((x, y)\) satisfying the condition \(x - y = (2n+1)\frac{\pi}{2}\), the domain of the function is: $$D(F) = \{(x, y) \in \mathbb{R}^2 | x - y \ne (2n+1)\frac{\pi}{2}, n \in \mathbb{Z}\}$$

Analyze the range of the function

Now, we will analyze the range of the function, i.e., the set of all possible output values. Since \(\tan^2(x)\) is non-negative and defined for all values of \(x\) except those of the form \((2n+1)\frac{\pi}{2}\), we can conclude that the range of the function $$F(x, y)=\tan^2(x-y)$$ is also non-negative.

Determine the range of the function

From the analysis in step 4, the range of the function is all non-negative real numbers. So, the range of the function is: $$R(F) = [0, +\infty)$$

Graph the function

We won't graph the function here, as it involves using a graphing utility. But we recommend you to use a graphing utility like Desmos, GeoGebra, or a graphing calculator to explore the graph of the function $$F(x, y)=\tan^2(x-y)$$. Make sure to adjust the window size and orientation as needed to get a better view of the surface. In this explanation, we have successfully determined the domain and range of the given function.

Key concepts

Domain and Range

When dealing with multivariable functions like \(F(x, y) = \tan^2(x-y)\), understanding the domain and range is crucial.

The **domain** refers to all the input values of \((x, y)\) for which the function is defined. For the tangent function, we need to avoid values where the tangent is undefined, which happens at odd multiples of \(\frac{\pi}{2}\). For \(F(x, y)\), this means avoiding values where \(x - y = (2n+1)\frac{\pi}{2}\), with \(n\) being any integer.

Therefore, the domain of this function is all real numbers \((x, y)\), except those that satisfy these conditions. It can be expressed as:

• \(D(F) = \{(x, y) \in \mathbb{R}^2 \ | \ x - y e (2n+1)\frac{\pi}{2}, n \in \mathbb{Z}\}\)

The **range** refers to all possible outputs of the function. Since \(\tan^2(x)\) is always non-negative, the range of \(F(x, y)\) is:

• \(R(F) = [0, +\infty)\)

These non-negative values are critical in understanding the bounds within which the function operates.

Tangent Function

The tangent function, \(\tan(x)\), is a fundamental function in trigonometry. It's important to understand its behavior, especially since it's involved in multivariable functions.

**Periodicity and Undefined Values**

The tangent function has a period of \(\pi\), meaning it repeats its values every \(\pi\) units. It is undefined at \((2n+1)\frac{\pi}{2}\), where \(n\) is an integer. These are the points where \(\tan(x)\) approaches infinity or negative infinity.

**Squared Tangent**

With \(F(x, y) = \tan^2(x-y)\), we square the tangent function, making it **always non-negative**. Squaring the function eliminates any negative values but still retains points where it is undefined. Hence it creates an interesting pattern that is important to note for both theoretical understanding and practical applications.

Graphing Utility

To visualize multivariable functions like \(F(x, y) = \tan^2(x-y)\), using a **graphing utility** is extremely helpful. These tools allow you to see the behavior and surface generated by the function.

**Choosing the Right Tool**

Popular graphing utilities including Desmos and GeoGebra can graph multivariable functions. They let you adjust the view, scale, and window size, providing flexibility in how you see the function's graph.

**Setting the Window**

While graphing \(F(x, y)\), it’s crucial to adjust the view to best represent the function’s critical points and behaviors. Since the tangent function has undefined points, being able to see how the function approaches these values is vital. This involves adjusting:

• **Orientation:** to view the best angles and important behavior at undefined points.
• **Zoom level:** to carefully observe the regions of interest.

Exploring this using graphing utilities enhances comprehension and confusion during manual graphing.

Mathematical Analysis

Mathematical analysis of multivariable functions like \(F(x, y) = \tan^2(x-y)\) involves breaking down the function into understandable components. This branch of mathematics provides tools for understanding limits, convergence, and behavior of functions.

**Critical Values and Excluded Points**

Start by focusing on where the function is undefined. In this case, as \(x - y\) approaches \((2n+1)\frac{\pi}{2}\), the function explodes towards infinity. It's crucial to identify and exclude these points from the domain using mathematical analysis.

**Function Behavior Over Domains**

Next, recognize how the rest of the domain behaves. For \(F(x, y)\), the function remains non-negative, ranging from zero to infinity. This perspective helps predict the model’s behavior and implement it practically.

Mathematical analysis thus helps in predicting outcomes and verifying theoretical models.