Problem 104 - Algebra and Trigonometry

Question

Without drawing a graph, describe the behavior of the graph of \(y=\cos ^{-1} x .\) Mention the function's domain and range in your description.

Step-by-Step Solution

Answer

The domain of the inverse cosine function \(y = \cos^{-1}x\) is \([-1, 1]\), and the range is \([0, \pi]\). The function is decreasing over its domain and symmetric with respect to the y-axis.

Step 1: Describe the Domain

The domain of the function \(y = \cos^{-1}x\) (also written as \(y = arccos(x)\)) consists of all the real numbers that can be input to the function, for which a real number output exists. The inverse cosine function is defined for all real numbers between -1 and 1, inclusive. So, the domain of \(y = \cos^{-1}x\) is \([-1, 1]\).

Step 2: Describe the Range

The range of the function consists of all the possible output values. The range of the inverse cosine function is \([0, \pi]\), because cosine values are between 0 and \(\pi\) in their natural domain.

Step 3: Describe the Behaviour

The inverse cosine function decreases as the input value increases over its domain. It starts at the upper y-bound of its range and ends at the lower y-bound. It also has the property of being an even function, meaning that it is symmetric with respect to the y-axis, i.e., \(\cos^{-1}(-x) = \cos^{-1}(x)\).