[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.
141. a. y=arccos(cos x)

Verified step by step guidance

1. Recall the definitions and ranges of the functions involved: the cosine function, \(\cos x\), has a domain of all real numbers and a range of \([-1, 1]\). The arccosine function, \(\arccos x\), is the inverse of cosine restricted to \([0, \pi]\), with domain \([-1, 1]\) and range \([0, \pi]\).
2. Identify the composite function: \(y = \arccos(\cos x)\). Since \(\cos x\) outputs values in \([-1, 1]\), which is the domain of \(\arccos\), the composite function is defined for all real \(x\). Thus, the domain of \(y\) is \((-\infty, \infty)\).
3. Determine the range of the composite function: Because \(\arccos\) outputs values only in \([0, \pi]\), the range of \(y\) is \([0, \pi]\). This means that even though \(x\) can be any real number, \(y\) will always be between \(0\) and \(\pi\).
4. Understand the behavior of the composite function: Since \(\arccos\) is the inverse of \(\cos\) only on \([0, \pi]\), for values of \(x\) outside this interval, \(\arccos(\cos x)\) will 'fold' the values back into \([0, \pi]\). This causes the function to be periodic and piecewise, reflecting the periodicity and symmetry of cosine.
5. When graphing \(y = \arccos(\cos x)\), expect a wave-like pattern that oscillates between \(0\) and \(\pi\), repeating every \(2\pi\). The graph will look like a series of 'V' shapes or 'triangles' due to the folding effect. This makes sense because the composition essentially maps all \(x\) values back into the principal range of \(\arccos\).

Key Concepts

Composite Functions

A composite function is formed when one function is applied to the result of another, written as (f ∘ g)(x) = f(g(x)). Understanding how to find the domain and range of composite functions requires analyzing the inner function's output and ensuring it fits within the domain of the outer function.

Inverse Trigonometric Functions

Inverse trig functions, like arccos, reverse the effect of their corresponding trig functions within a restricted domain. For arccos, the output range is [0, π], and it only accepts inputs in [-1, 1]. Recognizing these restrictions is key to determining the domain and range of compositions involving arccos.

Domain and Range of Trigonometric Compositions

When composing functions like y = arccos(cos x), the domain is all real numbers since cos x is defined everywhere, but the range is limited by arccos's output. The composition may simplify or behave differently over intervals due to periodicity and principal value restrictions, affecting the graph's shape and interpretation.