Problem 75 - Calculus Early Transcendentals

Question

Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. $$f(x, y)=\sin ^{-1}(x-y)^{2}.$$

Step-by-Step Solution

Answer

Question: Determine the domain of the function $$f(x, y)=\sin ^{-1}(x-y)^{2}$$. Answer: The domain of the function $$f(x, y)=\sin ^{-1}(x-y)^2$$ is given by the set of all ordered pairs \((x, y)\) such that \(y-1\leq x \leq y+1\). The region representing the domain can be visualized as an infinite strip, angled at \(45\) degrees.

Step 1: Identify the Input Function Range

First, we need to understand the range of the input function for the inverse sine function. The sine function has a range of \([-1,1]\), meaning the inverse sine function can only accept input values within this interval.

Step 2: Analyze the Given Function

The given function is $$f(x, y)=\sin ^{-1}(x-y)^{2}$$. The expression \((x-y)^{2}\) represents the input for the inverse sine function, and it must fall in the range of \([-1,1]\) as mentioned earlier.

Step 3: Solve the Inequality

Since \((x-y)^{2}\) must be between \(-1\) and \(1\), we can write the inequality as follows: $$-1\leq(x-y)^2\leq1$$ However, we notice that \((x-y)^2\) is always non-negative (a square is never negative), so there is no need to consider the left-hand side of the inequality. Hence, we have: $$(x-y)^2\leq1$$

Step 4: Solve for x and y

Now, to find the domain, we can write the inequality as follows: $$-(x-y)^2\leq0\leq(x-y)^2$$ Taking the square root, the inequality becomes: $$-1\leq x-y \leq1$$ We can rearrange this inequality to find values for \(x\) and \(y\): $$y-1\leq x \leq y+1$$

Step 5: Describe the Domain

The domain of the function $$f(x, y)=\sin ^{-1}(x-y)^2$$ consists of all the points \((x, y)\) satisfying \(y-1\leq x \leq y+1\). In words, the domain is comprised of all ordered pairs \((x, y)\) such that the value of \(x\) is between \(y-1\) and \(y+1\). This creates a region of allowed values for \(x\) and \(y\) that can be sketched as an infinite strip, angled at \(45\) degrees.

Step 6: Final Answer

The domain of the function $$f(x, y)=\sin ^{-1}(x-y)^2$$ is given by the set of all ordered pairs \((x, y)\) such that \(y-1\leq x \leq y+1\). The region representing the domain can be visualized as an infinite strip, angled at \(45\) degrees.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are critical in situations where we need to find an angle with a given trigonometric value. For example, if you know the sine of an angle is 0.5, the inverse sine function, also written as \(\sin^{-1}\), will give you the angle whose sine is 0.5.
The sine function naturally ranges from -1 to 1. This means that \(\sin^{-1}\) can only accept inputs within this range. If a value is outside this interval, it cannot be an input for \(\sin^{-1}\).
When we're analyzing functions that include inverse trigonometric components, it's essential to first confirm that the input to the inverse trigonometric function stays within the permissible range.

• This step involves identifying internal components of the function that contribute to the argument of the inverse function.
• In the case of \(\sin^{-1}((x-y)^2)\), we check to make sure that \((x-y)^2\) is within [-1, 1].

This enables us to determine the domain effectively and ensure correct solutions.

Inequalities

Inequalities are mathematical expressions that show the relationship of one side being lesser or greater than another. The process of solving inequalities often involves isolating the variable of interest to find the set of possible solutions it satisfies. For our context, say \((x-y)^2\), the basic understanding is required.
Key steps when dealing with inequalities include:

• Understanding the range or limits that dictate the validity of the function output.
• Modifying or rearranging the initial equation so it fits within required inequality bounds, such as rearranging \(x - y\).
  The inequality \(-1 \leq (x-y)^2 \leq 1\) quickly simplifies since squares are non-negative. Therefore, we narrowed it down to \( (x-y)^2 \leq 1\).

In essence, understanding and manipulating inequalities are essential to finding valid domains and ranges of functions.

Multivariable Functions

Multivariable functions incorporate more than one variable, such as \(f(x, y)\). Understanding these functions often includes determining how changes in one or more of the variables influence the function's outcome.
Specific to the function \(f(x, y)=\sin^{-1}((x-y)^2)\), the domain consists of all possible pairs \((x, y)\) where inequalities hold true. As such:

• The domain determines how both \(x\) and \(y\) relate to each other within specific intervals.
• Given the interval \(y-1 \leq x \leq y+1\), it means as \(y\) changes, \(x\) must lie within one unit distance to stay within the domain strip.

Understanding multivariable functions often involves visualizing their domain, which in this context is an infinite strip angled at 45 degrees. Here, the geometric approach can provide better insight into the interplay between variables in such functions.