Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The domain of the function \(f(x, y)=1-|x-y|\) is \(\\{(x, y): x \geq y\\}.\) b. The domain of the function \(Q=g(w, x, y, z)\) is a region in \(\mathbb{R}^{3}.\) c. All level curves of the plane \(z=2 x-3 y\) are lines.

Short answer

Provide an explanation or counterexample. A. The domain of the function f(x, y) = 1-|x-y| is equal to the set {(x, y): x ≥ y}. Answer: False Explanation: The domain of the function f(x, y) = 1-|x-y| covers the entire ℝ² space and is not restricted to only the set {(x, y): x ≥ y}. The counterexample provided shows that the function is also defined for x < y.

Step-by-step solution

a: Determine the domain of f(x, y) = 1-|x-y|

First, we need to find the domain of the function f(x, y) = 1-|x-y|. Since the function consists of a difference, there are no restrictions on x and y from the operation itself. Therefore, the domain covers the entire ℝ² space and it is not restricted to only the set {(x, y): x ≥ y}. So, the statement is not true. Counterexample: Take x = -1 and y = 1, where x < y. The function f(x, y) takes the value 1 - |-1 - 1| = 1 - 2 = -1, which is well-defined. Therefore, we can see that the function is also defined for x < y.

b: Determine the domain of Q = g(w, x, y, z)

The information given doesn't provide the definition of the function g(w, x, y, z), so we cannot state the domain without additional information. Therefore, we cannot confirm if the statement is true or false without the proper definition of the function.

c: Level curves of z = 2x-3y

Level curves are defined by setting the function equal to a constant value, in this case, z = 2x-3y = c, where c is any real number. If we rearrange the equation, we get y = ⅔x - ⅓c. The equation represents a family of lines with a slope of ⅔ and varying y-intercepts (-⅓c). Therefore, all level curves of the plane z=2x-3y are lines, and the statement is true.

Key concepts

Function Domain

In multivariable calculus, the domain of a function refers to all the possible input values that allow the function to be defined and real-valued. For a function of two variables, such as \(f(x, y)\), the domain is a subset of the two-dimensional real number space, \(\mathbb{R}^2\).

The domain includes all pairs of \((x, y)\) for which the function's formula yields a valid real number.

• Expressions like square roots or logarithms impose restrictions, such as requiring non-negative inputs for the square root or positive numbers for the logarithm.
• In the given example, the function \(f(x, y) = 1 - |x - y|\) has an absolute value operation but is not limited by domain constraints due to the subtraction and modularity.
• Consequently, the set \(\{(x, y): x \geq y\}\) does not restrict the domain of \(f\). Instead, the domain is all of \(\mathbb{R}^2\), since any real numbers \(x\) and \(y\) can be used.

Level Curves

Level curves help visualize functions of two variables by fixing the function at a certain constant value. By setting \(z = f(x, y) = c\), where \(c\) is a constant, the resulting expression describes the path or locus of points in the \((x, y)\)-plane that yield the same function value.

This idea simplifies analysis by reducing dimensions. In the given example, the equation \(z = 2x - 3y = c\) is explored.

• Rearranging this equation gives \(y = \frac{2}{3}x - \frac{c}{3}\).
• Each value of \(c\) defines a different line since the equation is linear in terms of \(x\) and \(y\).
• Varying \(c\) merely shifts the line vertically, resulting in parallel lines with a slope of \(\frac{2}{3}\) throughout the \((x, y)\)-plane.

Level curves like these help identify key features, such as symmetry and constraint interactions in higher-dimensional problems.

Real Number Space

Understanding real number spaces is crucial for grasping the concept of domains and level curves in multivariable functions. In mathematics, the real number space \(\mathbb{R}\) encompasses all the points on an infinite number line, representing real numbers.

• For a single variable function, \(\mathbb{R}^1\) is the domain, a straight line.
• As we extend functions to more variables, the dimensionality increases: \(\mathbb{R}^2\) represents pairs of numbers (like \(x, y\)), forming a plane.
• Similarly, \(\mathbb{R}^3\) involves triples (\(x, y, z\)), creating a 3-dimensional space, encapsulating the concept of volume or space.

Different dimensions of real number spaces allow for complex geometrical representations, which are foundational for understanding multivariable calculus. Spaces accommodate different concepts:
Domains are subsets of these real spaces indicating where functions remain valid.
Level curves show paths of consistent function values within these spaces, adding a layer of analysis to understand variations and patterns.