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Chapter 4: Problem 104

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.

Short Answer

Expert verified
The inverse cosine function \(y = cos^{-1}(x)\) maps any number from the domain \[-1, 1\] to an angle in the range \[0, \pi\]. It's a decreasing function: as \(x\) increases, \(y\) decreases.

Step by step solution

01

Understand the Inverse Cosine Function

The \(cos^{-1}(x)\) function, also known as arccosine, is the inverse function of the cosine function, restricted to the interval \([0, \pi]\). It returns the angle whose cosine is \(x\). In other words, if \(y = cos^{-1}(x)\), then \(cos(y) = x\). So it effectively 'undoes' the operation of the cosine function.
02

Identify the Domain

The domain of a function is the complete set of possible values of the independent variable. In the case of \(y = cos^{-1}(x)\), the domain is the set of all real numbers \(x\) such that \(-1 \leq x \leq 1\). These are the values for which the function is defined, because the cosine of an angle is always between -1 and 1.
03

Identify the Range

The range of a function is the complete set of possible resulting values of the dependent variable. For \(y = cos^{-1}(x)\), the range is the set of all real numbers \(y\) such that \(0 \leq y \leq \pi\). These are the values that the arccosine of any number can take.
04

Describe the Graph Behavior

The function \(y = cos^{-1}(x)\) is decreasing in its entire domain. It starts at \(y = \pi\) when \(x = -1\), and decreases to \(y = 0\) when \(x = 1\).

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