Problem 65 - Contemporary Precalculus

Question

Find a function \(f\) (other than the identity function) such that \((f \circ f \circ f)(x)=x\) for every \(x\) in the domain of \(f .\) [Several correct answers are possible.]

Step-by-Step Solution

Answer

Answer: One possible function is \(f(x) = \cos(x)\) for \(x\) in [−1, 1].

Step 1: Consider a common type of function

One type of function that often allows for function composition are trigonometric functions. Let's try composing the cosine function with itself to see if it fulfills the given condition.

Step 2: Compose the cosine function

First, we need to compose the cosine function with itself twice. \((f \circ f)(x) = \cos (\cos (x))\). Next, we need to compose this composition with the cosine function again: \((f \circ f \circ f)(x) =\cos (\cos (\cos (x)))\).

Step 3: Checking if our example satisfies the condition

To see if our example function \(\cos(x)\) satisfies the given condition, we can observe the properties of the cosine function. The cosine function is bounded between \([-1, 1]\). Only certain input values of the innermost cosine function pass through the [., .] range. If we restrict the domain of \(f(x)\) to \([-1, 1]\), the function will satisfy \((f \circ f \circ f)(x)=x\). It is necessary to mention that this result is approximate and near the fixed point where \(\cos(x)\) converges after repeated compositions.

Step 4: Present an example function

One possible function that fulfills the condition \((f \circ f \circ f)(x)=x\) within the restricted domain (and approximately outside) is: \(f(x) = \cos(x)\) for \(x\) in \([-1, 1]\). This function is by no means unique and other solutions may exist.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in the world of mathematics, often utilized to relate the angles of a triangle with the lengths of its sides, especially in right-angled triangles. But their application goes far beyond triangles; they play a crucial role in various fields like physics, engineering, and even music theory. Among the most commonly known trigonometric functions are sine, cosine, and tangent, each with a specific function domain and range.

The cosine function, denoted as \( \cos(x) \)—which we'll explore more shortly—is an example of a periodic function, meaning it repeats its values in regular intervals known as periods. This property makes the cosine function particularly interesting when looking at function compositions, as explored in the provided exercise.

Fixed Point

A fixed point in mathematics is a value that remains unchanged by a function. In other words, it's a point \( x \) such that \( f(x) = x \) for a given function \( f \). Fixed points are fascinating because they can indicate stability or signify a recurring condition within a system. For instance, when iterating a process, finding a fixed point could mean reaching a stable state that doesn't change with further iterations.

In the context of our exercise using cosine function, a fixed point would be critical in demonstrating that under certain conditions, the function \( f(x) = \cos(x) \) could satisfy \( (f \circ f \circ f)(x) = x \) within the domain where the property holds true. The discovery of such points can lead to more profound insights into the behavior of complex systems.

Cosine Function

The cosine function, symbolized as \( \cos \) and one of the primary trigonometric functions, describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It's defined for all real numbers when considering its graph on a unit circle and is inherently periodic with a period of \(2\pi\) radians (or 360 degrees).

The values of the cosine function range from -1 to 1, making it a bounded function. This characteristic is crucial in finding functions that compose with themselves to return their input, as in our exercise. When delving into function composition, it's helpful to visualize how the bounded range of cosine function affects the output and to consider the limitations it imposes on the domain of the composed functions.

Function Domain

The domain of a function consists of all the possible input values (often denoted as \( x \) values) that the function can accept without causing mathematical contradictions, such as division by zero or taking the square root of a negative number in the realm of real numbers. Determining the domain is a crucial step in understanding the behavior and limitations of a function.

When we talk about function composition, such as \( (f \circ f \circ f)(x) \) in our exercise, the function's domain must be considered carefully. It's because the output of one composition must fall within the domain of the subsequent function application. For the cosine function, restricting the domain to \( [-1, 1] \) may be necessary to ensure that the composition condition \( (f \circ f \circ f)(x) = x \) is met, as seen in the textbook exercise example.