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. $$p(x, y)=1-|x-1|+|y+1|$$

Short answer

Answer: The domain of the function is all real numbers for both \(x\) and \(y\), represented as \(\{(x, y) \in \mathbb{R}^2\}\), and the range is all real numbers greater than or equal to 1, represented as \([1, \infty)\). To graph the function, enter the function definition \(p(x, y) = 1 - |x-1| + |y+1|\) into a graphing utility's 3D graphing feature and adjust the window and orientation to get a better view of the function's behavior.

Step-by-step solution

Identify the domain

The domain of a function is the set of all potential input values (in this case, values of \(x\) and \(y\)) for which the function is defined. Since the function contains absolute value expressions, it is defined for all real numbers for both \(x\) and \(y\). Therefore, the domain of the function is all real numbers for both \(x\) and \(y\), which can be written as: $$ Domain = \{(x, y) \in \mathbb{R}^2\} $$

Identify the range

The range of a function is the set of all possible output values (in this case, values of \(p(x, y)\)) that result from valid input values from the domain. To determine the range, we must look at the function's behavior. Let's rewrite the function as: $$ p(x, y) = 1 - (|x - 1| - |y + 1|) $$ Notice that the minimum value occurs when \(|x - 1|\) and \(|y + 1|\) are equal, which leads to \(0\) for the expression inside the absolute value. In that case, \(p(x, y)\) will equal \(1\). Also, the function will increase when these two absolute values become more different. The maximum value of the function is not limited, so \(p(x, y)\) can take any real value greater than or equal to \(1\). Thus, the range of the function is: $$ Range = [1, \infty) $$

Graphing the function

To graph the function using a graphing utility, first enter the function definition \(p(x, y) = 1 - |x-1| + |y+1|\) into your preferred graphing utility's 3D graphing feature. Experiment with the window and orientation to give the best perspective on the surface. You may need to adjust the \(x\) and \(y\) axis ranges to include negative and positive values. This will help you get a better view of how the function behaves in the domain and range we determined above. In summary, after determining the domain and range and graphing the given function, we found that the domain consists of all real numbers for both \(x\) and \(y\), and the range is all real numbers greater than or equal to 1.

Key concepts

Absolute Value

Absolute value is a key concept when dealing with real numbers in mathematical functions. It refers to the magnitude of a number without considering its sign. For example, both \(-3\) and \(+3\) have the same absolute value, which is \(3\). This is because absolute value is essentially a measure of distance from zero on a number line.
In the function given \(p(x, y) = 1 - |x - 1| + |y + 1|\), the absolute value symbols play a crucial role. They transform any negative result from their subtraction operation within the symbols to a positive one. This affects the domain and range significantly because of the absolute value properties. The function can only give results equal to or greater than the computed base due to the non-negative nature of absolute values.
Here are some properties of absolute values that are important to remember:

• Absolute value is always non-negative.
• The absolute value of zero is zero.
• |a| = |−a|, meaning it treats numbers and their negatives the same way.
• |\(a+b\)| ≤ |a| + |b|; this is known as the triangle inequality.

Understanding how absolute values function is crucial when determining the range of functions like our example, where the absolute values directly influence the potential output values.

Real Numbers

Real numbers are the backbone of most mathematical functions and exercises because they encompass all possible numbers that one might encounter on a number line. This includes both rational numbers (like whole numbers, fractions) and irrational numbers (like \(\pi\) and square roots of non-perfect squares). Real numbers indicate both whole numbers and the infinite set of points in between any two whole numbers.
In the domain aspect of our exercise function \(p(x, y) = 1 - |x - 1| + |y + 1|\), real numbers are crucial. The function is defined for all real numbers for both \(x\) and \(y\) because the input can be any point on the plane. By understanding this, students will find why the domain is the set of all real number pairs \((x, y) \) in \(\mathbb{R}^2\).
Some important characteristics of real numbers are:

• The set of real numbers is continuous and contains no gaps.
• It is unbounded, extending infinitely in both positive and negative directions.
• Operations like addition, subtraction, multiplication, and division (except by zero) can be performed on real numbers.
• They can be ordered, meaning any two real numbers can be compared.

These properties make the set of real numbers the ideal candidate for defining domains in many algebraic functions.

Graphing Utility

Graphing utilities are incredibly useful tools for visualizing complex functions, especially those involving multiple variables like \(x\) and \(y\). These utilities enable students to see relationships and intersections among functions in a visual format, enhancing their understanding of abstract mathematical concepts.
When you handle the graph \(p(x, y) = 1 - |x-1| + |y+1|\), being familiar with a graphing utility is vital. By inputting this function into a graphing utility, you can create a 3D surface plot and explore its features more intuitively. By adjusting the viewing window and orientation, such software allows you to peer into the behavior and scope of a function's domain and range.
Here are ways a graphing utility can help:

• Provides dynamic interaction with the function, which can aid in better comprehension.
• Helps adjust the window for optimal views, showcasing critical points on a graph or surface.
• Can integrate with other computational tools to give analytical information about intersections, slopes, etc.
• Supports numerous visualization options such as plotting multiple functions simultaneously or showing the effect of transformations.

Thus, a graphing utility is an invaluable aid for learning and verifying complex functions and their characteristics.