Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Draw by hand the graph of each inverse function. $$y=\cot ^{-1} 2 x$$

Step-by-Step Solution

Answer

The graph of \( y = \cot^{-1}(2x) \) is a decreasing curve from \(\pi\) to 0 as \(x\) ranges from \(-\infty\) to \(\infty\).

Step 1: Understanding Inverse Functions

The given function is an inverse trigonometric function: \(y = \cot^{-1}(2x)\). This means that \(y\) is the angle whose cotangent is \(2x\). We need to understand the behavior of \(\cot^{-1}(x)\), which is defined for \(x \in \mathbb{R}\) and has a range of \((0, \pi)\).

Step 2: Determine Function Behavior

Investigate the function \(y = \cot^{-1}(2x)\): As \(x\) increases from \(-\infty\) to \(\infty\), \(2x\) also increases from \(-\infty\) to \(\infty\). Therefore, \(\cot^{-1}(2x)\) decreases from \(\pi\) to 0. This is a decreasing function.

Step 3: Analyzing Important Points

To plot the inverse function, identify key points. Evaluate: \( \cot^{-1}(0) = \frac{\pi}{2} \). To find when \( \cot^{-1}(2x) = \frac{\pi}{2} \), solve \(2x = 0 \) which gives \(x = 0\). Check limits: as \(x \to -\infty\), \(y \to \pi\); as \(x \to \infty\), \(y \to 0\).

Step 4: Sketch the Graph

Draw the graph: Start at the point (0, \(\frac{\pi}{2}\)), draw a decreasing curve which approaches \(\pi\) as \(x\) goes to negative infinity and approaches 0 as \(x\) goes to positive infinity. Mark these limits clearly on the vertical axis.

Key Concepts

Cotangent Function

The cotangent function is one of the six fundamental trigonometric functions, and it is the reciprocal of the tangent function. In simple terms, if the tangent of an angle in a right triangle represents the ratio between the opposite side and the adjacent side, the cotangent represents the ratio between the adjacent side and the opposite side. Thus, for any angle \( \theta \), the cotangent is expressed as \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)} \).
Inverse trigonometric functions, such as \( \cot^{-1}(x) \), are used to find the angle whose trigonometric ratio is given. So, when we refer to \( y = \cot^{-1}(2x) \), it signifies that \( y \) stands for the angle whose cotangent value is \( 2x \).
The function \( \cot^{-1}(x) \) is defined for all real numbers, \( x \in \mathbb{R} \), and yields angles within the range of \( (0, \pi) \). This range is essential because it denotes the principal values of the inverse cotangent function, providing a specific solution for any real number input.

Function Graphing

Function graphing involves visually representing mathematical functions on a coordinate system. This process helps us understand the behavior and properties of the function more easily.
When graphing \( y = \cot^{-1}(2x) \), we're mapping the behavior of this inverse function on a plane, with \( x \) representing the input and \( y \) representing the output angle.
To graph this function manually, it's helpful to identify significant points that illustrate the function's behavior and the range it covers.

• Identify the point where \( x = 0 \). Here \( y = \cot^{-1}(0) = \frac{\pi}{2} \).
• Notice that as \( x \to -\infty \), \( y \to \pi \), meaning the curve approaches \( \pi \) on the vertical axis.
• Conversely, as \( x \to \infty \), \( y \to 0 \), indicating the downward slope towards \( 0 \).

These points and limits help in sketching the general trend of the function as a decreasing curve.

Decreasing Functions

A decreasing function is one in which the function value decreases as the input value increases. In other words, for any two points \( x_1 \) and \( x_2 \) with \( x_1 < x_2 \), the function satisfies \( f(x_1) > f(x_2) \).
In the context of the inverse cotangent function \( y = \cot^{-1}(2x) \), as the input \( x \) increases, the value of the function decreases from \( \pi \) to \( 0 \). This gives the graph a distinct falling characteristic from left to right.
By incorporating numerical examples:

• Solve for \( x \) such that \( \cot^{-1}(2x) = \frac{\pi}{4} \), providing various values that demonstrate the function's decrease.
• Another confirmation: when \( x = -1 \), line up with the theoretical limit as \( x \to -\infty \), and \( y \to \pi \).

Understanding the behavior of decreasing functions helps in predicting future function values without graphing every point or calculation.

Trigonometric Functions

Trigonometric functions, including cotangent, sine, cosine, and more, are fundamental in mathematics. They model periodic phenomena and establish relationships between angle measures and side lengths in right triangles.
The concept of inverse trigonometric functions expands this idea, allowing you to determine angles based on known side ratios, hence enabling an extension from basic geometry to more sophisticated mathematical applications.
For instance, while \( \sin(\theta) \) gives a side ratio, \( \sin^{-1}(x) \) yields \( \theta \) itself. Similarly, \( \cot^{-1}(x) \) gives the angle whose cotangent is \( x \). This approach is invaluable in calculus and engineering where solutions must often involve angle measures rather than simple ratios.

• The study of these functions covers analyzing their domains (e.g., cotangent is undefined at multiples of \( \pi \)), ranges, and periodic nature.
• Knowing the properties of these functions and their inverses highlights symmetry, transformations, and graphical behavior, which are central to many fields in both pure and applied mathematics.
• Moreover, they reinforce the concept that trigonometric functions serve as periodic and angle-measuring tools across various scientific inquiries.