Problem 111 - Algebra and Trigonometry

Question

Graph each pair of functions in the same viewing rectangle. Use your knowledge of the domain and range for the inverse trigonometric function to select an appropriate viewing rectangle. How is the graph of the second equation in cach exercise related to the graph of the first equation? $$ y=\cos ^{-1} x \text { and } y=\cos ^{-1}(x-1) $$

Step-by-Step Solution

Answer

The graph of \(y=\cos^{-1}(x-1)\) compares to the graph of \(y=\cos^{-1}x\) as a translation one unit to the right. Selection of the viewing rectangle based on the domain and range of the functions is vital to correctly representing these relations.

Step 1: Understanding the function

First, consider the given functions. The functions are the inverse cosine, or arccosine, of x and x-1. The domain of the \(\cos^{-1}\) function is from -1 to 1, and the range is from 0 to \(\pi\). The \(\cos^{-1}(x-1)\) function is a horizontal shift of the \(\cos^{-1}x\) function.

Step 2: Defining the view rectangle

The view rectangle should be chosen based on the domain and range of both functions. The domain of \(\cos^{-1}(x-1)\) is between 0 and 2 (due to the shift of 1 in the function), and the range remains the same i.e. between 0 to \(\pi\). Therefore, the view rectangle should be chosen from 0 to 2 for x-values and 0 to \(\pi\) for y-values.

Step 3: Graphing the functions

Next, graph \(y=\cos^{-1}x\) and \(y=\cos^{-1}(x-1)\) in the same view rectangle. Both functions will have the shape of the \(\cos^{-1}\) function. However, the graph of \(y=\cos^{-1}(x-1)\) will be shifted 1 unit to the right of \(y=\cos^{-1}x\).

Step 4: Observing the relation between the graphs

Finally, observe how the graph of \(y=\cos^{-1}(x-1)\) relates to the graph of \(y=\cos^{-1}x\). It can be noted that the graph of \(y=\cos^{-1}(x-1)\) is just a shift or translation of \(y=\cos^{-1}x\) one unit to the right.

Key Concepts

Domain and Range

The domain and range are essential when working with inverse trigonometric functions. The domain determines all possible input values for a function, while the range specifies the set of possible outputs. For the inverse cosine function, denoted as \(\cos^{-1} x\), the domain is from \(-1\) to \(1\).
The range is from \(0\) to \(\pi\), meaning the output is always in radians covering this interval.

• Domain of \(\cos^{-1} x\): \([-1, 1]\)
• Range of \(\cos^{-1} x\): \([0, \pi]\)

Understanding these limits is crucial when setting graphs, ensuring they are positioned effectively within the viewing rectangle. Always adhere to the domain and range to select an appropriate portion of the graph to display.

Horizontal Shift

A horizontal shift modifies where a function appears on a graph without changing its shape. For the function \(y = \cos^{-1}(x-1)\), the entire graph of \(\cos^{-1} x\) shifts one unit to the right.
This shift alters the domain but keeps the range the same as the original function.

• Original domain of \(y = \cos^{-1} x\): \([-1, 1]\)
• Shifted domain of \(y = \cos^{-1}(x-1)\): \([0, 2]\)
• Range remains \([0, \pi]\)

Such transformations allow functions to be adjusted to better fit specific scenarios or problems, without altering the inherent characteristics of their graph.

Cosine Inverse Function

The cosine inverse function, known as \(\cos^{-1} x\), reverses the cosine function. While the cosine function takes an angle and returns a ratio, the inverse takes a ratio and outputs an angle within a specific range.
This function is vital in scenarios where you need angles instead of ratios.

• The output is always between \(0\) and \(\pi\), covering the upper semi-circle of a unit circle.

Comprehending how this inverse function behaves allows for precise angle calculations in various fields.
The shift in the function \(\cos^{-1}(x-1)\) shows flexibility in graph manipulation, revealing how the input substitution affects overall output while preserving the function's core properties.