Problem 54 - Calculus Early Transcendentals

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. $$g(x, y)=e^{-x y}.$$

Step-by-Step Solution

Answer

Answer: The domain of the function $$g(x, y)$$ is $$\{(x, y) | x \in \mathbb{R}, y \in \mathbb{R}\}$$ or $$(-\infty, \infty) \times (-\infty, \infty)$$. The range of the function is $$\{g(x, y) | 0 < g(x, y) \leq \infty\}$$ or $$(0, \infty)$$. To graph the function, use a graphing utility such as Desmos, GeoGebra, or a graphing calculator and adjust the window and orientation to get the best perspective of the surface.

Step 1: Find the Domain

The domain of a function is the set of all possible input values (x and y). In this case, since there is no restriction on the input values, the domain of $$g(x, y)$$ is: Domain: $$\{(x, y) | x \in \mathbb{R}, y \in \mathbb{R}\}$$ or in interval notation: Domain $$=(-\infty, \infty) \times (-\infty, \infty)$$.

Step 2: Find the Range

The range of a function is the set of all possible output values ($$g(x, y)$$). Since $$e^{u}$$ is always positive for any real $$u$$, in this case $$u = -xy$$, it can be any real number between 0 and infinity. So, the range of the function is: Range: $$\{g(x, y) | 0 < g(x, y) \leq \infty\}$$ or in interval notation: Range $$= (0, \infty)$$.

Step 3: Graph the Function Using a Graphing Utility

To graph the function $$g(x, y) = e^{-xy}$$, use a graphing utility such as Desmos, GeoGebra, or a graphing calculator. Enter the function and then adjust the window and orientation to get the best perspective of the surface. Summary: Domain: $$\{(x, y) | x \in \mathbb{R}, y \in \mathbb{R}\}$$ or Domain $$=(-\infty, \infty) \times (-\infty, \infty)$$ Range: $$\{g(x, y) | 0 < g(x, y) \leq \infty\}$$ or Range $$= (0, \infty)$$ Graphing the function will help visualize the domain and range, but remember that it is essential to understand the function and its properties to find these values analytically.

Key Concepts

Domain and Range

When working with multivariable functions like \(g(x, y) = e^{-xy}\), understanding the domain and range is crucial. The **domain** is all the possible input values for \(x\) and \(y\). For this function, both \(x\) and \(y\) can be any real numbers, i.e., there are no restrictions on them. This means the domain is all pairs \((x, y)\) where \(x \in \mathbb{R}\) and \(y \in \mathbb{R}\). In set notation, this is written as:

• Domain: \(\{(x, y) \mid x \in \mathbb{R}, y \in \mathbb{R}\}\)
• Interval notation: \((-\infty, \infty) \times (-\infty, \infty)\)

The **range** of a function is the set of possible output values. Because \(e^u\) (where \(u = -xy\)) is always positive for any input, the outputs of \(g(x, y)\) can be any real number greater than 0. The exponential function never actually reaches 0, but it can get very close, extending to infinity. Therefore, the range is all positive real numbers:

• Range: \(\{g(x, y) \mid 0 < g(x, y) \leq \infty\}\)
• Interval notation: \((0, \infty)\)

Graphing Utilities

Using graphing utilities can significantly aid in visualizing functions like \(g(x, y) = e^{-xy}\). These tools, such as Desmos or GeoGebra, enable you to graph complex surfaces, adjusting views for better comprehension. Here's how they can be helpful:

• **Input**: Easily enter the equation \(g(x, y) = e^{-xy}\).
• **Visualization**: Rotate and zoom into the graph to observe its behavior in 3D.
• **Exploration**: Experiment with different viewing windows to see various characteristics of the surface.
• **Interactivity**: Adjust parameters dynamically to see how changes affect the graph.

Graphing utilities provide an interactive way to understand the relationship between input values and the behavior of the function. They allow you to see how the surface emerges from the axes, making complex mathematical concepts more accessible.

Exponential Functions

The function \(g(x, y) = e^{-xy}\) is an example of an exponential function. Let's break down its key properties:Exponential functions in general have the form \(e^u\), where \(e\) is the base of natural logarithms, approximately equal to 2.718. These functions have several critical characteristics:

• **Always Positive**: No matter the value of \(u\), \(e^u\) is always positive.
• **Growth or Decay**: Depending on the sign of \(u\), they can represent exponential growth (positive \(u\)) or decay (negative \(u\)). In our case, \(u\) is \(-xy\), which typically signifies decay because of the negative coefficient.
• **Continuous and Smooth**: The graph of an exponential function is always a smooth curve without any breaks.
• **Asymptotic Behavior**: As \(x\) and \(y\) increase in opposite directions, \(g(x, y)\) approaches 0 but never reaches it.

Understanding how exponential functions behave can offer insights into how multivariable functions like \(g(x, y)\) perform over different regions of their domain. They play a significant role in mathematical modeling, often representing real-world phenomena such as population growth and radioactive decay.