Problem 18 - Calculus for Biology and Medicine

Question

In Problems \(15-24\), find the values of \(x \in \mathbf{R}\) for which the given functions are continuous. $$ f(x)=\cos (2 x) $$

Step-by-Step Solution

Answer

The function \( f(x) = \cos(2x) \) is continuous for all \( x \in \mathbb{R} \).

Step 1: Understand the Properties of Cosine

The cosine function, \( \cos(x) \), is continuous over all real numbers. Since it doesn’t have any breaks or gaps in its graph for any value of \( x \), we start by noting the continuity property of the cosine function.

Step 2: Apply Transformation Rules

The given function is \( f(x) = \cos(2x) \). Here, the argument of the cosine function is transformed by the factor of 2. Since this transformation is a composition of continuous functions (linear transformation and cosine function), the entire function \( \cos(2x) \) remains continuous for all real \( x \).

Step 3: Conclusion About Continuity

Since \( \cos(2x) \) is a composite of continuous functions (\(2x\) is continuous as a linear function and \(\cos(x)\) is continuous for all real numbers), the function \( f(x) = \cos(2x) \) is continuous across its entire domain, \( \mathbb{R} \).

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos(x) \), is one of the fundamental trigonometric functions. A crucial property of the cosine function is its continuity across all real numbers. This means, for every real number input, the cosine function returns a predictable and smooth output without any breaks or jumps.
To visualize this, imagine the graph of \( \cos(x) \). It moves in a smooth wave-like pattern as \( x \) increases or decreases, always remaining between -1 and 1. This smooth behavior is the hallmark of continuity.

• Periodic Nature: The cosine function repeats its values in regular intervals, every \( 2\pi \) radians. This periodic repetition contributes to its continuous nature.
• Boundedness: Since the cosine values are always between -1 and 1, the function remains well-behaved and continuous all over the real number line.

Function Composition

Function composition involves creating a new function by applying one function to the results of another. In mathematical terms, if you have functions \( h(x) \) and \( g(x) \), their composition is written as \( h(g(x)) \). This creates a "function of a function," chaining the operations of \( g \) and \( h \).
In our case, the function \( f(x) = \cos(2x) \) is a composition. Here’s why it matters:

• The linear function \( 2x \) is transformed first.
• Then \( \cos(x) \) is applied to the result of \( 2x \).

Composing functions often preserves continuity, especially if both \( h \) and \( g \) are continuous themselves. Since both \( 2x \) and \( \cos(x) \) are continuous functions, so remains their composition \( \cos(2x) \). This inclusion of one function into another demonstrates how mathematical operations can be combined smoothly and continuously, ensuring no disruptions in behavior.

Linear Transformation

A linear transformation is a type of function that maps input values to outputs using linear rules. In the most simple form, it might look like \( ax + b \), where \( a \) and \( b \) are constants. A linear transformation changes the input in a predictable way, often involving scaling (like multiplying by 2, in \( 2x \)), shifting, or rotating.
Here’s why linear transformations are essential in calculus:

• Simplicity: Linear transformations are easy to work with and understand, often serving as a building block for more complex functions.
• Continuity: A linear function like \( 2x \) is continuous because it doesn’t have any splits or jumps. It smoothly increases or decreases following a straight line.

In the function \( f(x) = \cos(2x) \), \( 2x \) represents a linear transformation that scales the input \( x \) by 2 before applying the cosine function. The seamless application of transformations like this helps ensure the overall function maintains its continuous property across the entire real line.