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Chapter 15: Problem 19

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

Short Answer

Expert verified
Answer: The domain of the function is $$\{(x, y) \in \mathbb{R} \times \mathbb{R} \setminus \{0\} \}$$, which means x can be any real number and y can be any real number except for 0.

Step by step solution

01

Identify the restrictions on the function

The given function is: $$f(x, y)=\sin \frac{x}{y}$$ The only possible restriction occurs when the denominator, y, is equal to zero, as division by zero is undefined. Therefore, y cannot be zero.
02

Find the domain

Since the only restriction on the function is that y cannot be zero, the domain of the function will be all real numbers for x and all real numbers for y except for 0. Thus, the domain of the function can be written as: Domain: $$ \{(x, y) \in \mathbb{R} \times \mathbb{R} \setminus \{0\} \} $$ This indicates that x can be any real number and y can be any real number except for 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Restrictions on Functions
When working with functions, it's essential to identify any restrictions that might affect their domain. The domain of a function refers to the set of all possible input values (x) that the function can accept. Restrictions usually arise due to operations within the function that are not universally defined or permissible. These can include operations like square roots of negative numbers, logarithms of non-positive numbers, and most importantly, division by zero.

In the given exercise, we are working with a function that involves a fraction as its input to a trigonometric function 鈥� \(f(x, y) = \sin \frac{x}{y}\). Here, the primary restriction to consider is the denominator, which must never be zero. When identifying domains, always look for such restrictions by examining the denominator and considering what values make an expression undefined.
Implications of Division by Zero
Division by zero is a crucial concept in mathematics that leads to undefined results. This aspect dramatically influences the domain of a function. When a fraction has a denominator of zero, it attempts to divide a number by nothing, which is mathematically impossible. Therefore, any function featuring a fraction must have a provision stating that the denominator cannot be zero.

In our function \(f(x, y) = \sin \frac{x}{y}\), the fraction \(\frac{x}{y}\) becomes undefined if y equals zero. Therefore, we exclude y = 0 from the domain. It's important to always perform this check because permitting y to equal zero would result in an undefined expression, which is not allowed in any sound mathematical context.
Basics of Trigonometric Functions
Trigonometric functions, such as sine, cosine, and tangent, are functions of an angle and are fundamental in understanding periodic phenomena. The function in our example, \(\sin \frac{x}{y}\), uses the sine function. Sine measures the y-coordinate of a point on the unit circle, where the angle is the input.Although sine itself can accept any real number as an input, restrictions can come from the other operations performed with its inputs, in this case, the fraction \(\frac{x}{y}\). This doesn't impose any trigonometric-specific constraints per se but is guided by the rule regarding division by zero.In summary, while the sine function is generally unrestricted, the calculations that precede it (especially involving possible division by zero) significantly impact the domain. Hence, always consider both the trigonometric function and the operations applied to its input when identifying domain restrictions.

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Most popular questions from this chapter

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Steiner's problem for three points Given three distinct noncollinear points \(A, B,\) and \(C\) in the plane, find the point \(P\) in the plane such that the sum of the distances \(|A P|+|B P|+|C P|\) is a minimum. Here is how to proceed with three points, assuming the triangle formed by the three points has no angle greater than \(2 \pi / 3\left(120^{\circ}\right)\) a. Assume the coordinates of the three given points are \(A\left(x_{1}, y_{1}\right)\) \(B\left(x_{2}, y_{2}\right),\) and \(C\left(x_{3}, y_{3}\right) .\) Let \(d_{1}(x, y)\) be the distance between \(A\left(x_{1}, y_{1}\right)\) and a variable point \(P(x, y) .\) Compute the gradient of \(d_{1}\) and show that it is a unit vector pointing along the line between the two points. b. Define \(d_{2}\) and \(d_{3}\) in a similar way and show that \(\nabla d_{2}\) and \(\nabla d_{3}\) are also unit vectors in the direction of the line between the two points. c. The goal is to minimize \(f(x, y)=d_{1}+d_{2}+d_{3}\) Show that the condition \(f_{x}=f_{y}=0\) implies that \(\nabla d_{1}+\nabla d_{2}+\nabla d_{3}=0\) d. Explain why part (c) implies that the optimal point \(P\) has the property that the three line segments \(A P, B P\), and \(C P\) all intersect symmetrically in angles of \(2 \pi / 3\) e. What is the optimal solution if one of the angles in the triangle is greater than \(2 \pi / 3\) (just draw a picture)? f. Estimate the Steiner point for the three points (0,0),(0,1) and (2,0)
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