Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Write short answers and fill in the blanks. Consider the inverse tangent function, \(y=\tan ^{-1} x\) or \(y=\arctan x\) (a) What is its domain? (b) What is its range? (c) For this function, as \(x\) increases, \(y\) increases. Therefore, it is a(n) _____ function. (d) Is there any real number \(x\) for which arctan \(x\) is not defined? If so, what is it (or what are they)?

Step-by-Step Solution

Answer

(a) Domain: all real numbers. (b) Range: \((-\frac{\pi}{2}, \frac{\pi}{2})\). (c) Increasing. (d) No real number is undefined for arctan \(x\).

Step 1: Understand the Inverse Tangent Function

The inverse tangent or arctangent function, denoted as \( y = \tan^{-1} x \) or \( y = \arctan x \), gives the angle whose tangent is \( x \). It "inverts" the tangent function.

Step 2: Identify the Domain of arctan(x)

The domain of the arctangent function \( y = \arctan x \) is the set of all real numbers, \( \mathbb{R} \). This means that for any real number \( x \), \( \arctan x \) is defined.

Step 3: Identify the Range of arctan(x)

The range of the arctangent function \( y = \arctan x \) is the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). This is because the arctangent yields angles that are between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), exclusive.

Step 4: Determine if arctan(x) is Increasing or Decreasing

The arctangent function \( y = \arctan x \) is an increasing function because as \( x \) increases, \( y \) also increases. This is true for every \( x \) in its domain.

Step 5: Check for Undefined Values of arctan(x)

The function \( y = \arctan x \) is defined for all real numbers \( x \). Therefore, there is no real number \( x \) for which \( \arctan x \) is not defined.

Key Concepts

Domain and Range

The domain and range are crucial concepts for understanding inverse trigonometric functions like the arctangent. The **domain** of a function is the complete set of input values for which the function is defined. For the inverse tangent, denoted as \(y = \arctan x\), the domain is quite simple: it includes all real numbers. This means you can plug any real number into \(\arctan x\) and find a valid angle as a result.

Now, let's talk about the **range**. The range of a function refers to all possible outputs it can produce. For \(y = \arctan x\), the range is the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\). This means the function will only give angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), but not including the endpoints. These are the angles you would find on a unit circle between the specified intervals, representing all possible outputs for \(\arctan x\).

• Domain: All real numbers \(\mathbb{R}\)
• Range: \((-\frac{\pi}{2}, \frac{\pi}{2})\)

Function Behavior

The behavior of a function describes how it acts or changes depending on the values of its input. For the inverse tangent function, \(y = \arctan x\), the behavior is straightforward – it is an **increasing function**. This means that as you increase the input value \(x\), the output value \(y\) also increases. Such behavior is typical for inverse tangent due to its smooth and continuous curve that rises steadily from one horizontal asymptote to the other.

This is why we describe \(\arctan x\) as a monotonic function, specifically a monotonic increasing function. Being increasing is a defining feature of \(\arctan x\), ensuring that no matter what real number you input, the function will consistently produce a larger output as compared to smaller input values. This can be visually observed on the graph of \(\arctan x\) as a consistent upward trend."

Real Numbers

Real numbers are fundamental in mathematics, representing points on an infinite number line. They include both rational numbers, like \(2.5\), and irrational numbers, such as \(\pi\), without exception or gap. When discussing inverse trigonometric functions like \(y = \arctan x\), real numbers play a critical role.

The domain of \(\arctan x\) - all real numbers, signifies that you can input any real number into this function without encountering undefined values or limits. Such functions are built to handle the comprehensive and inclusive range of real numbers, making them broadly applicable in numerous mathematical contexts.

Real numbers are integral in describing the domain because they cover all values that \(\arctan x\) can accept, ensuring the function is always valid and meaningful for any number you choose. There are no exclusions or discontinuities in the set of real numbers for this function.

• Real numbers include every number that can be found on the number line.
• The domain of \(\arctan x\) effortlessly encompasses all these numbers.