Problem 6 - Calculus Early Transcendentals

Question

Express \(Q(x)=\cos ^{4}\left(x^{2}+1\right)\) as the composition of three functions; that is, identify \(f, g,\) and \(h\) so that \(Q(x)=f(g(h(x))).\)

Step-by-Step Solution

Answer

Answer: The functions f, g, and h are: - \(f(x) = x^4\) - \(g(x) = \cos(x)\) - \(h(x) = x^2 + 1\)

Step 1: Identify the innermost function h(x)

The innermost function appears in the brackets, which is \(x^2 + 1\). Thus, let \(h(x) = x^2 + 1\).

Step 2: Identify the middle function g(x)

The middle function is the cosine function that operates on \(h(x)\). Thus, let \(g(x) = \cos(x).\)

Step 3: Identify the outermost function f(x)

The outermost function takes the result of \(g(h(x))\) (which is \(\cos(x^2 + 1)\)) and raises it to the power of 4. Thus, let \(f(x) = x^4\).

Step 4: Combine the three functions to express Q(x)

Now that we have identified \(f, g,\) and \(h\), we can write \(Q(x)\) as a composition of these functions: \(Q(x) = f(g(h(x))) = f(g(x^2 + 1)) = f(\cos(x^2 + 1)) = \cos^4(x^2 + 1)\). The functions f, g, and h are: - \(f(x) = x^4\) - \(g(x) = \cos(x)\) - \(h(x) = x^2 + 1\)

Key Concepts

Functions

Functions are fundamental building blocks in mathematics that allow us to map inputs to outputs in a precise way. A function consists of three primary components:

• **Input:** Also known as the domain, these are the values that you put into the function.
• **Rule:** This is the operation or series of operations that the function performs on the input.
• **Output:** Also known as the codomain, this is the result obtained after applying the rule to the input.

Functions can be expressed using variable notation, such as \( f(x) \), where \( x \) is the input variable. The value of the function \( f \) at \( x \) is what the function outputs. Functions can be composed, meaning the output of one function becomes the input to another. This composition allows for the creation of complex expressions using simpler fundamental functions.

Cosine Function

The cosine function is a trigonometric function, commonly denoted as \( \cos(x) \). It's typically used to find the horizontal coordinate corresponding to an angle on the unit circle. Cosine is important in various fields, including physics and engineering, for describing periodic phenomena such as waves.
The cosine function has the following properties:

• It is periodic with a period of \( 2\pi \).
• It ranges between -1 and 1 for all real inputs.
• It is even, meaning \( \cos(-x) = \cos(x) \).

In function composition, the cosine function often comes into play when needing to apply trigonometric transformations after an algebraic one. In the example given, it forms the middle layer of the function composition \( Q(x) = \cos^4(x^2 + 1) \), where it follows the computation of \( h(x) = x^2 + 1 \).

Power Function

Power functions are a type of function where the variable is raised to a constant power. These functions are commonly written as \( x^n \) where \( n \) is a constant exponent.
Key characteristics of power functions include:

• The exponent \( n \) determines the shape of the graph.
• If \( n \) is positive, as in \( x^4 \), the function grows rapidly and is positive for positive \( x \) values.
• They can be even or odd functions. For instance, \( x^2 \) is even, while \( x^3 \) is odd.

Power functions appear frequently in modeling natural phenomena, often indicating relationships of growth. In the expression \( Q(x) = \cos^4(x^2 + 1) \), the power function \( f(x) = x^4 \) is the outermost function in the composition, indicating a transformation applied to the entire cosine function after it has processed \( h(x) \). This emphasizes the effect of the power on the oscillating output of the cosine function.