Problem 34 - Calculus Early Transcendentals

Question

\(29-38\) Determine the set of points at which the function is continuous. $$G(x, y)=\tan ^{-1}\left((x+y)^{-2}\right)$$

Step-by-Step Solution

Answer

The function is continuous at all points except where \(x+y=0\).

Step 1: Understanding the Function

The function given is \( G(x, y) = \tan^{-1}((x+y)^{-2}) \). We identify it as the arctangent of \((x+y)^{-2}\). The function \(\tan^{-1}(z)\) is continuous for all real numbers \(z\). This means \(G(x, y)\) will be continuous at any point \((x, y)\) where \((x + y)^{-2}\) is defined.

Step 2: Analyze Domain Restrictions

For the function \((x+y)^{-2}\) to be defined, \(x+y\) cannot be zero. This is because you cannot divide by zero in mathematics. Therefore, the expression \((x+y)^{-2}\) is undefined when \(x+y=0\).

Step 3: Determine Points of Continuity

Based on Step 2, the function \(G(x, y)\) is continuous at all points \((x, y)\) except where \(x+y=0\). This line \(x+y=0\) represents a line of discontinuity in the xy-plane. Hence, \(G(x, y)\) is continuous for all points except those on this line.

Key Concepts

Arctangent Function

The arctangent function, often denoted as \( \tan^{-1}(z) \) or \( \arctan(z) \), is the inverse of the tangent function. It maps values from the tangent function back to angles, primarily focusing on output values within the interval \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \). This means whenever you input a real number \( z \) into the arctangent function, it returns an angle or value in radians within this specified range.

One of the key properties of \( \tan^{-1}(z) \) is that it is continuous for all real numbers. This is crucial because it means that the arctangent function will smoothly connect points without breaks or jumps. So, when you look at visual representations or graphs of this function, it's a smooth curve beyond any discontinuity in the input expressions (like \( x+y \) in the provided exercise).

Keep in mind, while the arctangent function is always continuous, the continuity of composite functions like \( G(x, y) = \tan^{-1}((x+y)^{-2}) \) depends on other factors, particularly the input expression \( (x+y)^{-2} \). Always ensure that any expressions involved are well-defined.

Domain Restrictions

Domain restrictions arise when certain operations within functions are not defined for some values. In the context of the exercise, we have the function \( (x+y)^{-2} \).

• Firstly, to calculate \( (x+y)^{-2} \), \( x+y \) must not be zero since any number to the power of \(-2\) translates to \( \frac{1}{(x+y)^2} \). Division by zero is undefined in mathematics.
• The domain of \( (x+y)^{-2} \) therefore omits points where \( x+y = 0 \).

Understanding these restrictions helps identify potential issues in continuity for composite functions. In this case, \( (x+y)^{-2} \) is defined and continuous everywhere except along the line \( x + y = 0 \). This insight is fundamental as it directly affects where the overall function \( G(x, y) \) can be continuous.

Discontinuity Points

In any mathematical function, points of discontinuity are where the function cannot be defined or transitions abruptly instead of smoothly. For \( G(x, y) = \tan^{-1}((x+y)^{-2}) \), the primary concern for discontinuity is derived from the expression \( (x+y)^{-2} \).

Here's how it unfolds:

• The function \( (x+y)^{-2} \) is undefined at \( x+y = 0 \) because it leads to division by zero. This creates a discontinuity line in the xy-plane, represented by \( x+y=0 \), which translates to a straight line passing through the origin with a slope of \(-1\).
• Consequently, \( G(x, y) \) inherits this discontinuity from \( (x+y)^{-2} \). In other words, wherever \( (x+y)^{-2} \) cannot be evaluated, \( G(x, y) \) will be discontinuous.

Recognizing discontinuity points is essential, especially when analyzing behavior and application of functions in mathematical and real-world contexts. When assessing a function's continuity, remember to analyze the contributing expressions for potential undefined scenarios.