Problem 15 - Calculus Early Transcendentals

Question

Find the domain of the following functions. $$f(x, y)=\sin \frac{x}{y}.$$

Step-by-Step Solution

Answer

Answer: The domain of the function is the set of all ordered pairs \((x, y)\) of real numbers where \(y\) is nonzero: $$\text{Domain}(f) = \{(x, y) \in \mathbb{R}^2 | y \neq 0\}.$$

Step 1: Identify potential issues

Here, the only potential issue we can see is the division by \(y\), which would render the function undefined if \(y = 0.\) Other than that, there doesn't seem to be any case where the function would be either undefined or non-real. So, we only need to avoid having \(y = 0.\)

Step 2: Determine possible input values

Since the only issue we find is having \(y = 0,\) the domain of this function is the set of all ordered pairs \((x, y)\) where \(y\) is nonzero: $$\text{Domain}(f) = \{(x, y) \in \mathbb{R}^2 | y \neq 0\}$$

Step 3: Describe the domain fully

We have found the domain of the function as $$\text{Domain}(f) = \{(x, y) \in \mathbb{R}^2 | y \neq 0\}.$$ This means that the function \(f(x, y)\) is defined for every pair \((x, y)\) of real numbers, except when \(y\) equals zero.

Key Concepts

Multivariable Functions

Multivariable functions involve multiple input variables, unlike single-variable functions which only consider one. In the case of the function given in the exercise, \(f(x, y)=\sin \frac{x}{y}\), we deal with two variables: \(x\) and \(y\). Whether you're in physics, engineering, or mathematics, multivariable functions play a crucial role. They model real-life phenomena where outputs depend on various factors. Understanding them helps to depict systems with more comprehensive detail.
These types of functions are denoted typically as \(f(x_1, x_2, \, \ldots, x_n)\), where the function decides an output based on these various inputs. When analyzing such functions, one significant step is identifying conditions that these variables must follow for the function to be defined, which leads us to consider the domain.

Undefined Expressions

Undefined expressions in mathematics arise when you perform arithmetic operations that don't produce meaningful results. A common example is division by zero. When analyzing \(f(x, y)=\sin \frac{x}{y}\), the potential problem is the denominator \(y\) becoming zero.
Division by zero is undefined because it contradicts the basic properties of numbers. Imagine trying to distribute a number among zero parts—it simply doesn’t make sense.
Therefore, when examining a function, it is crucial to determine where these undefined scenarios can occur. Ensuring such values are removed from the domain allows us to keep our function well-defined, working smoothly in every mathematic procedure involving it.

Real Numbers Domain

The concept of a domain is tied closely to the function's input values, essentially representing all possible \((x, y)\) pairs we can plug into the function. For real numbers, the domain includes all these possible values, except those which make the function undefined.
For \(f(x, y) = \sin \frac{x}{y}\), the real number domain consists of all ordered pairs \((x, y)\) such that \(y eq 0\). Removing undefined expressions like \(y = 0\) ensures that the calculation \(\frac{x}{y}\) remains valid.
Identifying the real numbers domain involves:

• Pinpointing variables' problematic values.
• Excluding these values, ensuring the function remains defined at every point within its domain.

Understanding this concept enhances our ability to interpret and solve mathematical exercises involving more than a single variable.